OPTIMIZATION AND APPLICATIONS · Vagif Abdullayev1 1 Cybernetics Institute of ANAS, Baku, Azerbaijan; [email protected] Let us consider the following optimal control problem for

University of Evora, Portugal

Institution of Russian Academy of SciencesDorodnicyn Computing Centre of RAS

III International Conference on Optimization

Methods and Applications

OPTIMIZATION

AND

APPLICATIONS

(OPTIMA-2012)

Costa da Caparica, Portugal, September 2012

PROCEEDINGS

Moscow — 2012

UDC 519.658

Proceedings include extended abstracts of reports presented at theIII International Conference on Optimization Methods and Applications"Optimization and applications" (OPTIMA-2012)held in Costa da Caparica, Portugal, September 23–30, 2012.Edited by V.I. Zubov.

ISBN 978–5–91601–051–0

Научное издание

c© Федеральное государственное бюджетное учреждение наукиВычислительный центр им. А. А. Дородницына

Российской академии наук, 2012

Organizing Committee

Vladimir Bushenkov, Chair, University of Evora, PortugalYury G. Evtushenko, Chair, Dorodnicyn Computing Centre of RAS,Russia

Alexander P. Afanasiev, Institute for System Analysis of RAS, RussiaRasim Alguliev, Institute of Information Technology of ANAS,AzerbaijanAnatoly S. Antipin, Dorodnicyn Computing Centre of RAS, RussiaOleg Burdakov, Linkoping University, SwedenVladimir A. Garanzha, Dorodnicyn Computing Centre of RAS, RussiaAlexander I. Golikov, Dorodnicyn Computing Centre of RAS, RussiaVladimir Goncharov, University of Evora, PortugalAlexander Yu. Gornov, Institute System Dynamics and Control Theory,SB of RAS, RussiaMiloica Jacimovic, Montenegrin Academy of Sciences and Arts,MontenegroAlexander V. Lotov, Dorodnicyn Computing Centre of RAS, RussiaYuri Nesterov, CORE Universite Catholique de Louvain, BelgiumVera Roshchina, University of Evora, PortugalYaroslav D. Sergeyev, University of Nizhni Novgorod, Russia, Universityof Calabria, ItalyGueorgui Smirnov, University of Minho, PortugalTatiana Tchemisova, University of Aveiro, PortugalIvetta A. Zonn, Dorodnicyn Computing Centre of RAS, RussiaVladimir I. Zubov, Dorodnicyn Computing Centre of RAS, Russia

Contents

Vagif Abdullayev Numerical solution to optimal control problems for

loaded dynamic systems with integral conditions . . . . . . . . . . . . . . . . . . . . . . . 10

Alexander P. Abramov On second eigenvalue in Leontiev’s model . . . 13

Oleg V. Abramov Optimization Techniques for Parametric Synthesis of

Engineering Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Alexander Afanasyev, Elena Putilina Maximizing the volume of

three-dimensional bodies on the basis submetric transformation . . . . . . . . . 21

Kamil Aida-zade, Yegana Ashrafova Optimal control problems with-

out initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Z. Akbari, R. Yousefpour, M. R. Peyghami New Nonsmooth Trust

Region Method for Unconstraint Locally Lipschitz Optimization Problems 25

Keyvan Amini, Masoud Ahookhosh, Somayeh Bahrami A Conju-

gate Gradient Projection Algorithm for systems of Large-Scale Nonlinear

Monotone Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Anton Anikin Software implementation of an algorithm for finding the

optimal control using a graphics accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Maxim Anop, Yaroslava Katueva The reduction of the optimal para-

metric synthesis to the linear programming problem . . . . . . . . . . . . . . . . . . . . 34

Anatoly Antipin Boundary value games in optimal control . . . . . . . . . . . 36

Dmitry Arkhipov, Alexander Lazarev, Elena Musatova Minimiza-

tion of maximum lateness for M stations with tree topology . . . . . . . . . . . . 42

Armen Beklaryan Existence theorems for elliptic equations in un-

bounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Vladimir Bushenkov, Bento Caldeira, Georgi Smirnov On the de-

termination of the earthquake slip distribution via linear programming

techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Miguel Constantino, Xenia Klimentova, Ana Viana New Integer

Programming formulations for the Kidney Exchange Problem . . . . . . . . . . . 54

V.V. Dikusar, E.S. Zasukhina Identification of parameters in model of

water transfer in soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5

Anna Dorjieva Improvement Technology for the Accuracy of Solution of

Unconstrained Argument Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Olga Druzhinina, Natalia Petrova On optimal stabilization with re-

spect to a part of variables for multiply connected controlled systems . . . 67

W. Dunin-Barkowski, L. Vyshinskiy Numerical experiments on com-

puter model of a cerebellum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Yu.G. Evtushenko, A.A. Tretyakov P -th order methods for solving

nonlinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Evgeny R. Gafarov, Alexandre Dolgui, Alexander Lazarev Some

Complexity Results for the Simple Assembly Line Balancing Problem . . . 81

Shamil Galiev, Maria Lisafina, Vitalii Yudin Optimization of a mul-

tiple covering of a surface taking into account its relief . . . . . . . . . . . . . . . . . 86

Alexander Gasnikov, Eugenia Gasnikova Stochastic subgradient

barrier-multiplicative descent for entropy optimization . . . . . . . . . . . . . . . . . . 91

Dinh Thanh Giang, Phan Thanh An, Le Hong Trang An Efficient

Algorithm for Determining the Lower Convex Hull of a Finite Point Set in

3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Alexander I. Golikov LP projection algorithm and Newton method for

solving dual LP problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Evgenij Golshtejn Many-Person Games With Convex Structure . . . . . . 99

Vladimir Goncharov, Fatima Pereira Proximal Analysis and Regu-

larity of Viscosity Solution to some Hamilton-Jacobi Equation . . . . . . . . . . 101

Aleksander Gornov Multimethod’s algorithm for parametric identifica-

tion of nonlinear dynamic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Anna D. Guerman Optimization Problems in Astrodynamics . . . . . . . . 106

Nguyen Ngoc Hai, Phan Thanh An Blaschke Convergence Theorem

for G-type Convex Sets in Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Niyaz Ismagilov, Farit Nasyrov Pathwise optimal control of diffusion

type processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

M. Jacimovic, I. Krnic On accuracy of the regularization method of

constrained ill-posed quadratic minimization problems . . . . . . . . . . . . . . . . . . 115

6

Ruben V. Khachaturov Cubes Lattice’s properties investigation and

possibilities of its application in Combinatorial Optimization . . . . . . . . . . . 118

Vladimir R. Khachaturov General Theory of Optimization on Finite

Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Michael Khachay, Maria Poberii Hyperplane Covering Problems.

Complexity and Approximation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Elena Khoroshilova Leontief’s model as a boundary value problem in

optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Konstantin Kobylkin Approximation to minimum committee problem

for system of linear inequalities in IR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Pavel Korenev, Alexander Lazarev Metric for the total tardiness min-

imization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Olga Kostyukova, Tatiana Tchemisova New CQ-free optimality cri-

terion for convex SIP problems with polyhedral index sets . . . . . . . . . . . . . . 139

Vladimir Krivonozhko, Finn Førsund, Andrey Lychev Optimiza-

tion methods for measurement of returns to scale in the non-radial DEA

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Ulyana Kulbida, Olga Kaneva Selection of the target audience by the

leverage method in the expert system for advertising specialist . . . . . . . . . 147

Samir Kuliev Zonal control of lumped systems on different classes of

feedback functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Maya Laskova, Alexander Lazarev, Elena Musatova The Heuristic

Approach to movement optimization on single-track part of the railway net 156

Alexander V. Lotov, Georgij Kamenev Finite time-interval robust-

ness study of dynamic systems with imprecisely identified parameters . . . 159

A.A. Lukovenko, T.M. Tikhomirova Estimation of economic damage

from human mortality by external causes on macro-, meso-, micro- levels 162

Eloısa Macedo, Adelaide Freitas Statistical Methods and Optimization

in Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Igor E. Mikhailov, L.A. Muravey On control with coefficients for high

order partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7

Arsalan Mizhidon Analytic design of an optimal controller under per-

manent stochastic disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Boris Mordukhovich Optimal control of the sweeping process . . . . . . . . 178

Evgenii Murashkin Optimal deformation during the creep . . . . . . . . . . . 178

L.A. Muravey, V.M. Petrov, A.M. Romanenkov Modeling and op-

timization of ion-beam etching process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Elena Musatova, Alexander Lazarev, Nail Husnullin Special algo-

rithm for Three-Stations Railway problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Daniar Nurseitov, Maksim Shishlenin, Syrym Kasenov Numerical

solution of two-dimensional inverse problem for the Helmgoltz equation . 192

Nataliya Obrosova, Alexander Shananin Production model in the

conditions of unstable demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Andrei Orlov, Sergei Pinigin Global search in bilinear separation prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Valeriy Parkhomenko Ensemble calculations application for estimation

and optimization of climate model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Sergey Perzhabinsky, Valery Zorkaltsev Interior point algorithms . 207

Lev F. Petrov Interactive optimization as a tool for finding the complex

periodic solutions in nonlinear dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

M.R. Peyghami, H. Tavakoli, M. Ahmadian Attari On the Semidef-

inite Representation of the Maximum Optimal Rate Problem in LDPC

Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Alexander Plakhov Problems of optimal resistance in Newtonian aero-

dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Boris Polyak L1 problems in control and numerical methods for their

solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Mikhail Posypkin, Izrael Sigal Object-oriented Framework for Dy-

namic Control of the Parallel Branch-and-Bound . . . . . . . . . . . . . . . . . . . . . . . 221

Ekaterina Rassadnikova, Aida Valeeva, Nelli Magafurzyanova

Graf of decision logistics making for problem of goods delivery . . . . . . . . . . 223

Nataliya Sedova Discontinuous control and Lyapunov functions for non-

linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

8

Simon Serovajsky Optimal control of nonlinear parabolic equations and

the differentiability of the control-state mapping . . . . . . . . . . . . . . . . . . . . . . . 229

Iliyas Shakenov Inverse problems for parabolic equations with infinite

horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Kanat Shakenov Solution of the parametric inverse problem of stochastic

optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

O. Shcherbina, A. Sviridenko Graph-based local elimination algo-

rithms for sparse discrete optimization problems . . . . . . . . . . . . . . . . . . . . . . . 240

Yuri N. Sotskov, Omid Gholami, Frank Werner Heuristic Algo-

rithms for a Job-Shop Problem with Minimizing Total Job Tardiness . . . 245

A.A. Tretyakov P -regular nonlinear optimization. High order optimality

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Evgeniya A. Vorontsova Separating plane algorithm with additional

clipping for convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

V.P. Vrzheshch, N.P. Pilnik, I.G. Pospelov Equilibrium Model of

the Russian Economy for the period of Global Financial Crisis . . . . . . . . . . 256

R. Yousefpour A Modified Steepest Descent Method Based on BFGS

Method for Locally Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Vitaly Zhadan The primal affine-scaling method for semidefinite pro-

gramming with steepest descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Valery Zorkaltsev The point projections on linear manifold . . . . . . . . . . . 267

Vladimir Zubov, Alla Albu, Andrey Albu The effect of the setup

parameters on the evolution of the substance crystallization process . . . . . 270

Anna Zykina, Nikolay Melenchuk Convergence of the two-step extra-

gradient method in a finite number of iterations . . . . . . . . . . . . . . . . . . . . . . . . 274

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

9

Numerical solution to optimal control problems forloaded dynamic systems with integral conditions

Vagif Abdullayev1

1 Cybernetics Institute of ANAS, Baku, Azerbaijan; vaqif [email protected]

Let us consider the following optimal control problem for the processdescribed by a system, which is linear with respect to the phase variable,of loaded ordinary differential equations:

x(t) = A(t, u)x(t) +

l3∑

s=1

Bs(t)x(

t s) + C(t, u) , t ∈ (t0, T ], (1)

where x(t) ∈ En is the phase variable; u(t) ∈ U ⊂ Er is the controlvector-function from the class of piecewise continuous functions, admissi-ble values of which belong to the given compact set U ; the (n×n)matrixfunctions A(t, u)–6=const, Bs(t), s = 1, ..., l3, and n−dimensional vector-function C(t, u) are continuous with respect to t and continuously differen-

tiable with respect to u. The points of loading time

t s ∈ [t0, T ] ,

t s+1 >

t s, s = 1, 2, ..., l3 are given.Nonseparated multipoint and integral conditions are given in the fol-

lowing form:

l1∑

i=1

t2i∫

t2i−1

Di(τ )x(τ)dτ +

l2∑

j=1

Djx(tj) +

l3∑

s=1

Dsx(

t s) = L0, (2)

where the continuous matrix function Di (τ) and scalar matrices Dj ,

Ds

have the dimension (n× n); L0 is the n-dimensional vector; ti,tj are thepoints of time belonging to [t0, T ]; ti+1 > ti, tj+1 > tj , i = 1, ..., 2l1 −1, j = 1, ..., l2 − 1, l1, l2,l3 are given.

To be definite, without loss of generality, make an assumption that

min(t1, t1

)= t0, max

(t2l1 , tl2

)= T, (3)

and for all i = 1, ..., 2l1, j = 1, ..., l2, s = 1, ..., l3, the following conditionholds

10

tj ,

t s∈ [t2i−1, t2i] . (4)

The target functional is as follows:

J(u) = Φ(x(t)) +

T∫

t0

f0(x, u, t)dt→ minu(t)∈U

, (5)

where the function Φ is continuous with respect to its arguments alongwith the private derivatives, and f0(x, u, t) is continuously differentiablewith respect to (x, u), and continuous with respect to t; t = (t1, t2, ..., t2l1+l2)is the ordered union of points of the sets t = (t1, t2, ..., tl2), t = (t1, t2, ..., t2l1)

and

t = (

t 1,

t 2, ...,

t s), i.e. tj < tj+1, j = 1, ..., 2l1 + l2 + l3 − 1.Suppose that the problem (1) and (2) is solvable under any admissible

control u(t) ∈ U ∈ Er.Theorem. The gradient of the functional in the problem (1)-(5) is

determined as follows:

(grad J(u))∗ =∂f0(x, u, t)

∂u(t)− ψ∗(t)

[∂A∗(t, u)

∂u(t)x(t) +

∂C∗(t, u)

∂u(t)

]. (6)

where the vector-function ψ(t) ∈ En and the vector λ ∈ En satisfy thefollowing differential equation:

ψ(t) = −A∗(t, u)ψ(t)−l3∑

s=1

δ(t−

t s

) T∫

t0

Bs∗(t)ψ(t)dt+

+

l1∑

i=1

[χ(t2i)− χ(t2i−1)] D∗(t)λ+

∂f0∗(x, u, t)

∂x(t), (7)

the following boundary conditions

ψ(t0) =

(∂Φ(x(t))

∂x(t1)

)∗+ D∗

1λ, for t0 = t1 ,(∂Φ(x(t))∂x(t1)

)∗, for t0 = t1 ,

(8)

ψ(T ) =

−(∂Φ(x(t))

∂x(tl2)

)∗− D∗

l2λ, for tl2 = T ,

−(∂Φ(x(t))∂x(t2l1 )

)∗, for t2l1 = T,

(9)

11

the following jump conditions at the intermediate points tj, for whicht0 < tj < T ,

ψ+(tj)− ψ−(tj) =

(∂Φ(x(t))

∂x(tj)

)∗+ D∗

jλ, j = 1, ... l2, (10)

the following jump conditions at the loading points

t s, for which t0 <

t s < T,

ψ+(

t s)− ψ−(

t s) =

(∂Φ(x(t))

∂x(

t s)

)∗

+

D∗sλ, s = 1, ... l3, (11)

and the following jump conditions at the points ti, i = 1, ..., 2 l1, forwhich t0 < ti < T,

ψ+(ti)− ψ−(ti) =

(∂Φ(x(t))

∂x(ti)

)∗

, i = 1, ... 2l1. (12)

Here “*” is the transposition sign; δ(·) is the delta function; χ(t) isthe Heaviside function.

For numerical solution to the problem, we propose to use standardprocedures of first order optimization. To determine the value of thegradient by the formula (6), at each iteration, it is necessary to: 1) solvethe problem (1) under the current control with multipoint and integralconditions (3) using the technique of convolving integral conditions intolocal conditions (here we mean to use the results of the work [1]); 2) solvethe adjoint problem (7)-(11) using the generalized operation of iteratedshifts, making a special emphasis on the participation of the parameters λin the conditions (10)-(11) (here we mean to use the results of the works[2, 3]). Following the shift of the conditions, we obtain an algebraic systemof equations with the n(l3 + 2) unknowns λ, with the values of the phasetrajectory at one of two ends of the interval, and with the loading points[4].

Results of numerical experiments obtained by solving the problems of

12

the form (1)-(5) are given in the presentation.

References

1. K.R. Aida-zade, V.M. Abdullayev. “Numerical solution to differential equa-tion systems involving nonseparated point and integral conditions”, Proceed-ings of the High Technical Educational Institutions of Azerbaijan, Informaticsand Automatic Control, 13, No. 4, 64–70 (2011).

2. K.R. Aida-zade, V.M. Abdullayev. “Numerical Solution of Optimal ControlProblems with Nonseparated Conditions on Phase State”, Appl. and Com-put. Math. An International Journal, 4, No. 2, 165–177 (2005).

3. V.M. Abdullayev, K.R. Aida-zade. “Numerical solution of optimal controlproblems for loaded lumped parameter systems”, Computational Mathemat-ics and Mathematical Physics, 46, No. 9, 1566–1581 (2006).

4. K.R. Aida-zade. “ A Numerical method of restroring the parameters of adynamic system”, Kibern. Sistemn. Anal., 40, No. 3, 392–399 (2004).

On second eigenvalue in Leontiev’s model

Alexander P. Abramov1

1 Computing Center RAS, Moscow, Russia; [email protected]

Consider the Leontiev simple dynamic model. In our case, the econ-omy is closed and consists of n sectors. The output vector x(t) ∈ Rn

satisfies the inequality Y x(t) ≤ x(t − 1) at step t, t = 1, 2, . . ., whereY = yij is a technological matrix. Assume that the matrix Y is primi-tive.

If we put in order the eigenvalues of the matrix Y , we get

|λ1| 6 |λ2| 6 . . . 6 |λn−1| < λn = λY ,

where λY is the Frobenius eigenvalue of the matrix Y .Denote by pY and by xY the left and the right Frobenius vectors of

the matrix Y such that pY xY = 1. Using these vectors, let us definethe square matrix L such that L = xY pY . Recall [1] that the sequence(Y/λY )t, t = 1, 2, . . . tends to L as t → ∞. In this case the degree ofconvergence is estimated by the formula

∥∥∥(Y/λY )t − L∥∥∥l∞

< Crt, (1)

13

where r is any number such that |λn−1|/λY < r < 1 and C = C(r, Y ) issome positive constant.

In [2] it was considered the model of decentralised economy with Leon-tiev’s technologies such that the economic system may asymptoticallyreach the balanced growth. Denote by xp(1) the vector of the planned out-put at step 1. According to the model, for the normed sequence of output,we get (Y/λY )txp(1). This sequence tends to ϑxY , where ϑ ≡ pY xp(1).

Using (1), we get∥∥∥(Y/λY )t xp(1)− ϑxY

∥∥∥∞< Crt, (2)

where C is some positive constant.In our case, the scalar r in (1)-(2) is bounded below by |λn−1|/λY . This

bound depends on second eigenvalue λn−1. Let us consider the economiccontents of this value. Further assume that the matrix Y is positive.

Recall that the elements of a positive matrix comply with Hopf’sbound [1], so that

|λn−1|λY

6M − µM + µ

< 1, (3)

where M = maxi,j

yij and µ = mini,j

yij . This estimate shows that the

decreasing of the range (M−µ) decreases the upper bound for the fraction|λn−1|/λY .

Assume that λn−1 > 0. Denote by z some eigenvector correspondingto λn−1 such that

Y z = λn−1z. (4)

Since the matrix Y is irreducible it follows that the vector z has thecomponents of different signs. Let xk be a component such that zk < 0.

We say that the technological process is the reverse one if this pro-cess extracts the resources from the final product. It is assumed thatthe volumes of these resources are exactly the same as the original in-put. Moreover, the consumer properties are identical for the recoveredresources and for the normal products. We stress that the definition ofthe reverse process is speculative.

Using this definition, we see that |zk| on the right-hand side in (4) isequal to the input for the reverse process from some external source. Atthe same time the product yikzk is equal to the volume of the resource oftype i coming into the system as a result of the reverse process.

14

On the other hand, the negative value of the sum∑i ykizi means

that the amount of resource of type k, which will be produced in allreverse process, exceeds the consumption of this product in all ”direct”processes. Thus, the sector k such that zk < 0, may be considered as amulti-product producer of resources. This producer consumes only theproduct of type k, which comes into the system from an external source.We stress that the consumption of these products should grow with thesame rate as the rate of economic growth corresponding to the turnpike.In addition, a reverse process should occur immediately at the beginningof each step such that the released resources were available for using inall starting ”direct” processes.

It is clear that the additional resources increase the rate of balancedgrowth in sectors with ”direct” processes. This fact is expressed formallyby the inequality (1/λn−1) > (1/λY ). It should be stressed that theeconomic system can not be closed if it uses at least one reverse process.

It follows from bound (3) that the ratio λn−1/λY tends to zero asthe parameter µ tends to M from below. It means that the decreasingof the range (M − µ) increases the rate of balanced growth compared tothe 1/λY , which can be achieved by the reversion of some technologicalprocesses.

This conclusion is obvious because the system may use some resourceswithout limit from an external source, and the specific consumption ofthese resources is approximately the same in all sectors.

In addition, the high efficiency of the reverse processes in terms ofincreasing the balanced growth rate also positive effects on the rate ofconvergence of the normalized outputs (see (2)).

The author was supported by the Russian Foundation for Basic Research

(project no. 11-07-00201).

References

1. R.A. Horn, C.R. Johnson. Matrix Analysis, Cambridge University Press,Cambridge (1986).

2. A.P. Abramov. Balanced Growth in Models of Decentralized Economy, LI-BROKOM Publishers, Moscow (2011). (in Russian)

15

Optimization Techniques for Parametric Synthesis ofEngineering Systems

Oleg V. Abramov1

1 Institute for Automation and Control Processes, Far Eastern Branch RAS,

Vladivostok, Russia; [email protected]

The synthesis of engineering systems consists of two basic parts: de-veloping of structure (structural synthesis) and internal parameter valueschoosing (parametric synthesis). This paper proposes the approach andsome algorithms for seeking a numerical solution of the parametric opti-mization problem (parametric synthesis) of analogous electronic circuit.The circuit design optimization process is confounded by three significantkinds of unfavorable complexity, namely– the complexity of function and gradient evaluation, which can be ex-treme,– the combinatorial complexity of approximation algorithms which arebasically exponential in n, the dimension of the design parameter space,– the uncertainty of models used for analogous electronic circuits, and interms of the statistical uncertainty of the values assumed by the param-eters of these models. In parametric optimization, the topology of thecircuit and component types are fixed.

In general the optimal parametric synthesis problem can be stated asfollows [1].

Suppose that we have a circuit which depends on a set of n parametersx = (x1, . . . , xn). We will say that circuit is acceptable if Y(x) satisfy theconditions (1):

a ≤ Y(x) ≤ b, (1)

where Y(x), a and b are m-vectors of circuit responses (output param-eters) and their specifications. The inequalities (1) define a region Dx inthe space of design parameters

Dx = x | a ≤ Y(x) ≤ b (2)

Dx is called the tolerance margin domain (region of acceptability) for thecircuit. It is region in the input parameters space.

Let given the characteristics of random processes X(t) of system pa-rameters variations, a region of admissible deviation - Dx and a service

16

time T , find such a deterministic vector of parameter ratings (nominals)xr = (x1r , . . . , xnr) that the probability

Pr(xr, T ) = Pr [X1(t)− x1r, . . . , Xn(t)− xnr ] ∈ Dx, ∀t ∈ [0, T ]be maximized.

Any optimization technique requires, first, a method of objective func-tion calculation and, second, an extremum searching method which allowsto find a solution with a minimum cost.

The practical algorithm of the stochastic criterion calculation is basedon the conventional Monte Carlo method. The Monte Carlo method ap-proximates Pr(xr , T ) by the ratio of number of acceptable realizations(falling in region Dx) – Na to the total number of trials – N . Unfortu-nately, often the regionDx is unknown. It is given only implicitly throughsystem’s equations and the systems response functions. If we do not knowthe region Dx, then a Monte Carlo evaluation of probability Pr(xr, T ) atparticular nominal value xr requires N system analysis for each trial setof parameter xr. Typically, hundreds of trials are required to obtain areasonable estimate for Pr(xr, T ). Optimization requires the evaluationof our probability Pr(xr, T ) for many different values of the nominal val-ues of the parameters xr. Therefore to make practical the use of MonteCarlo techniques in statistical system design, it is necessary to reduce thenumber of system analysis required during optimization.

As a solution, the following two-steps technique and the correspondingalgorithms can be used for practical reliability optimization.

The first step consists in replacing the original stochastic criterionwith a certain deterministic one, allowing nearby optimum solutions tobe obtained. The two such objective functions are possible. One of themis a so-called a ”minimal serviceability reserve” that can be presented inthe general form:

F (x) = mini−1,m

[(ai − Yi(x)/wi − 1],

where Yi(x) – the i–th output value, ai – the i-th constraint (Y (x) ≤a) and wi – the i-th weight coefficient. From this, we have a followingoptimization problem:

xr = argmaxx∈D

F (x).

It means that such a nominal point should be found that would havethe largest distance from the acceptability region margins.

17

An other method, which can be used, for the reliability optimizationis so-called ”equal densities method”. This method is of combined type,which uses statistical data and a deterministic optimization technique.

At the first step we should estimate distribution density function(DDF) for output parameter. As can be shown analytically, probabil-ity maximum will be achieved, if DDF will be shifted such, that bothlower and upper constraints will cut equal densities on DDF [1].

Now the first design step is completed. At the same time the nextdesign step must be made if the reliability index that was achieved byusing deterministic methods is not high enough. This step is a directprobability optimization, i.e.; methods of stochastic optimization shouldbe used here.

It should be pointed out that most of optimization methods havethe highest convergence speed when they start at a ”good” initial point.Therefore, it would be most natural to get a previous solution as an initialpoint for the next design step.

Particularly effective way to decrease total design time on the phase ofmodrelling and statistical optimization is to use modern supercomputingtechnologies and parallel processing techniques [2].

Note that evaluation of exstrPr(xr, T ) requires a global optimization.The simplest method of global optimization is scanning (full enumeration)method. However, such method is considered computationally inefficient.The effective way to decrease optimization time is to use modern super-computing technologies and parallel algorithms.

The nominal values of the schematic components xn commonly usedfor engineering systems should lie in the predefined set of values as itis required by various standards and technical recommendations, it issometimes more preferable to search the optimal vector inside the discreteset of values that conforms to the standards and lies in the acceptableregion Dx.

Let us have the known internal parameters vector xr ∈ Dx. Thereforeat the each point of discrete set Din

r =xinr /xr ∈ Dx

we need to find

the Pr(xinr ) estimate. The optimum nominal vector xr we are looking for

can be found as a solution of the following task

xr = maxxr

Pr(xinr ) (3)

In the simplest case the solution can be found by complete check of each

18

element of the set Dinr with the probability estimation for each of them.

The set Dinr building can be implemented as a preliminary procedure that

puts the element values to the database. The optimum search process canbe performed in parallel mode. This algorithm can be presented as a two-level distributed process that requires RN processors for implementation(here R means the number of elements in the set Din

r ).Note an analogous method would apply to the general optimization

problem by using the regionalization (discretization) approach. Regional-ization consists of dividing the tolerance box into a finite number of nonoverlapping regions DJ , to form a grid. Then, the center or midpoint cjof each region Dj , is chosen to ”represent” entire region. The informationon a variation of values of internal parameters can be presented as limitsof their values, i.e.

ximin ≤ xi ≤ ximax, i = 1, n (4)

The area in space of internal parameters assigned by relations (4), repre-sents n-dimensional the orthogonal parallelepiped, which we shall nameas a beam of tolerances B∂ :

B∂ = x|xi min ≤ xi ≤ ximax, i = 1, n

It is possible to define the area of acceptable values of parameters Dx

by methods based on multivariate exploration of tolerance region B∂ . Atmultivariate exploration a beam (region) of tolerances B∂ can be repre-sented by a finite number of sampling points.It is obvious, that in situations the discrete change of all parameters si-multaneously is taken into account, and set of incompatible situations issampled representation of a beam B∂ . Each of situations is some samplingpoint representing appropriate subregion of a tolerance box (quantum-neighborhood).For each of R possible situations output parameters Y(x) is computed,condition (1) is tested and discrete set of parameter nominals Din

r =xinr /xr ∈ Dx

is formed.

The optimum nominal vector xr we are looking for can be found as a solu-tion of the task (3). A second method of using parallel parallel processingtechniques to maximize reliability is random search method.On the basis of the proposed parallel methods and algorithms for regionof acceptability location, modeling and discrete optimization a computer-

19

aided reliability-oriented distributed design (CARD) system has been de-veloped [3]. The CARD system builds mathematical models and calcu-lates ratings of component parameters so that achieve the highest preci-sion, acceptability (manufacturing yield) or reliability of analog electroniccircuits under design. The CARD system includes:— the simulation module (it facilitates the use of a variety of simulationprograms for electronic circuits design);— the module for deterministic and statistical analysis;— the module for objective function (reliability and/or manufacturingyield) calculation;— the optimization module.The system is organized from group of computers incorporated in a net-work. Such system allows using all advantages of client - server technol-ogy.CARD system uses a widely distributed PSPICE 9.0 circuit simulationprogram that allows simulating a large class of analogous devices in di-rect current, frequency and time domains.

References

1. O.V.Abramov. Reliability-directed parametric synthesis of stochastic systems.Nauka, 1992.

2. O.V.Abramov. ”Parallel algorithms for computing and optimizing reliabilitywith respect to gradual failures”, Automation and Remote Control, vol. 71,No. 7, 1394-1402, 2010.

3. O.V.Abramov, Y.V. Katueva, D.A. Nazarov. ”Distributed computing envi-ronment for reliability-oriented design”, Reliability and Risk Analysis: The-ory and Applications, vol. 2, No. 1, 39-46, 2009.

20

Maximizing the volume of three-dimensional bodieson the basis submetric transformation

Alexander Afanasyev1, Elena Putilina2

1 Institute for Systems Analysis RAS, Moscow, Russia; [email protected] Institute for Systems Analysis RAS, Moscow, Russia; [email protected]

The classical problem of maximizing the volume of three-dimensionalbodies on the basis submetric transformation reduces to the of optimalcontrol problem. Consider the following cases: a bodies of revolution,cylinders, convex polyhedra.

Optimal control problems without initial conditions

Kamil Aida-zade1, Yegana Ashrafova2

1 Azerbaijan State Oil Academy, Baku, Azerbaijan;

kamil [email protected] Cybernetics Institute of ANAS, Baku, Azerbaijan; y [email protected]

One of the most important classes of problems of distribution of bound-ary regimes is the class of “problems without initial conditions”. If controlof boundary regimes lasts long enough, then due to the friction inherentin any real physical system, the influence of initial data on the process’sbehavior subsides with the course of time. Thus we naturally come to aproblem without initial conditions.

Tikhonov A.N. was the first to study boundary-value problems with-out initial conditions for parabolic and hyperbolic equations in his work[1]. He gave the method of investigating problems without initial condi-tions,as well as their first rigorous solution [2]. In the well-known work[3], he investigated uniqueness of the solution to problems without initialconditions as applied to the heat conduction equation (Fourier problems).

In the present work, we investigate an optimal control problem withoutinitial conditions, considering, as an example, the wave process arising in

21

hydrocarbon raw material pipeline transportation systems; we also inves-tigate an optimal control problem for the heat conduction process withoutinitial conditions.

Let the process be described by the following hyperbolic differentialequation system [4]:

−∂P (x,t)

∂x = ∂Q(x,t)∂t + aQ(x, t), t ∈ [0, T ], x ∈ [0, l] ,

−∂P (x,t)∂t = c2 ∂Q(x,t)

∂x ,

(1)

P (0, t) = u0(t), P (l, t) = ul(t), (2)

where a is the friction coefficient; c is the velocity of sound in the environ-ment; (P (x, t), Q(x, t)) is the process phase state, determined from thesolution to the system (1)–(2) under the corresponding admissible valueof the optimizable control vector-function u = (u0(t), ul(t)).

Suppose that there are constraints, proceeding from technological con-ditions and technical requirements, on the control vector-functions, of theform

u ≤ u(t) ≤ u, t ∈ [0, T ], (3)

Determining the process initial state be long to some admissible set ofpairs of functions D = Q0(x), P0(x) : x ∈ [0, l], for each of which all theconditions of existence and uniqueness of the solution to the correspondingboundary-value problem are fulfilled.

The objective of the problem is to find such values of the boundarycontrols u1(t), u2(t), t ∈ (0, T ],under which the following functional:

J(u) =1

mesD

∫

D

l∫

0

[[Q(x, T ;u,Q0, P0)− qT (x)]2+

+[P (x, T ;u,Q0, P0)− pT (x)]2]ρ(Q0)ρ(P0)dxdQ0dP0 → min (4)

takes its minimal value. Here (Q(x, T ;u,Q0, P0), P (x, T ;u,Q0, P0))is thesolution to the initial boundary-value problem(1), (2), and (4) for somechosen admissible initial conditionsQ0(x), P0(x). The functional (4) de-termines the mean value of the deviation of the process state at t = T

22

from the given desired state (qT (x), pT (x)) for all possible initial condi-tions (Q0(x), P0(x)) ∈ D;ρ(Q0), ρ(P0) are the density functions of thedistribution of the initial values on the set D. The time interval [t0, T ],on which the process state does not depend on the values of the initialconditions given at t = 0, plays one of the major roles in investigation ofthe optimal control and boundary-value problems.

We can use the method of variation of the optimizable parameters toobtain the formulas for the gradient of the functional [5].

Let ψi(x, t) = ψi(x, t;u,Q0, P0), i = 1, 2 be the solution to the nextadjoint initial boundary-value problem:

−∂ψ1(x,t)

∂x = ∂ψ2(x,t)∂t , x ∈ (0, l), t ∈ (0.T ),

−∂ψ1(x,t)∂t = c2 ∂ψ2(x,t)

∂x − aψ1(x, t),

(5)

ψ1(x, T ) = 2[Q(x, T )− qT (x)],ψ2(x, T ) = 2[P (x, T )− pT (x)], x ∈ (0, l), (6)

ψ2(0, t) = 0, t ∈ [0, T ], ψ2(l, t) = 0, t ∈ [0, T ]. (7)

Here (P (x, T ) = P (x, T ;u,Q0, P0), Q(x, T ) = Q(x, T ;u,Q0, P0)) is thesolution to the initial boundary-value problem (1), (2), and (4) under anyadmissible u = u(t), Q0 = Q0(x), P0 = P0(x).

The formulas for the components of the gradient of the target func-tional with respect to the control functions u0(t), ul(t) are determined inthe following form:

gradu0(t)J = − 1

mesD

∫

D

ψ1(0, t)ρ(Q0)ρ(P0)dQ0dP0, t ∈ [0, T ], (8)

gradul(t)J =1

mesD

∫

D

ψ1(l, t)ρ(Q0)ρ(P0)dQ0dP0, t ∈ [0, T ]. (9)

For numerical solution to the optimal control problem in distributedsystems (1)-(5),we propose to use first order iterative optimization meth-ods based on the application of the analytical formulas derived for the

23

gradient of the target functional with respect to the optimizable func-tions. For example, we can make use of gradient projection methods:

uk+1 = PrU(uk − λkgradJ(uk)), k = 0, 1, ...,

or conjugate gradient projection methods [5]. Here u0 = [u00(t), u0l (t)] is

some given initial value of the control; gradJ(u) is the gradient of thetarget functional with respect to the optimizable vector-functions; λk isthe step of one-dimensional search in the line of the anti-gradient of thetarget functional; PrU (·) is the projection operator (this operator has asimple form for the positional constraints (3) [5]).

The formulas for the gradient of the target functional obtained abovecan also be used to formulate necessary optimality conditions (in the formof maximum principle in the variation form).

In the work, we will also consider the one-dimensional problem ofoptimal boundary control of the heating process without initial conditions.Formulas for the gradient of the target functional in this problem will begiven. Results of numerical experiments of the solution to the optimalcontrol problems will be given at the presentation.

References

1. A.N. Tkhonov, A.A. Samarski. The equations of mathematicalphysics,Nauka, Moscow, (1977).

2. G.O. Vafodorova. “The problems without initial conditions for one no classicequation ”, Diff. Equations, 39, No. 2, 278–280, (2003).

3. A.N. Tkhonov. “The uniqueness theorems for heat equations. Mathematicalcollection ”, Mathematical collection, 42, No. 2, 199–216, (1935).

4. I.A. Charniy. he transient motion of real fluid in the pipelines, Nedra,Moscow, (1975).

5. F.P. Vasilyev. Optimization methods, Factorial Press, Moscow, (2002).

6. O.. Ladijenskaya. The boundary problems of mathematical physics, Nauka,Moscow, (1973).

24

New Nonsmooth Trust Region Method forUnconstraint Locally Lipschitz Optimization

Problems

Z. Akbari1, R. Yousefpour2, M. R. Peyghami3

1 Department of Mathematics, K.N. Toosi University of Technology, Tehran,

Iran; z [email protected] Department Mathematical Sciences, University of Mazandaran, Babolsar,

Iran; [email protected] Department of Mathematics, K.N. Toosi University of Technology, Tehran,

Iran; [email protected]

Abstract

In this paper, a local model is presented for the locally Lipschitzfunctions. This local model is constructed by an approximation ofthe steepest descent direction. The steepest descent direction isan element of ǫ-subdifferential with minimal norm. In fact in thequadratic model, gradient is replaced by an approximation of thesteepest descent direction. The classical trust region method is ap-plied on this model. We prove that this algorithm is convergent byusing the bounded positive definite matrices. The positive definitematrix is updated in each iterations by the BFGS method. Finally,the presented algorithm is implemented by MATLAB code.Keywords: Trust region, Lipschitz functions, Local model, Stei-haug method

Introduction

The nonsmooth unconstraint minimization problem is one of the impor-tant problems in the real world. For example in smooth case, the penaltyand lagrangian functions are nonsmooth optimization problems. Also,these problems are used in control optimization. Therefore, solving theseproblems are attended.

The trust region (TR) method is an iterative method. In this method,the objective function is trusted by a local model. In each iteration, themodel is reduced instead of objective function in the adequate region. If

25

f : Rn → R is continuously differentiable, then the local model is definedas follows

m(xk, Bk)(p) = f(xk) +∇f(xk)T p+ 1/2pTBkp, (1)

where Bk is adequately selected. If f is twice continuously differentiable,then Bk is the Hessian matrix. In some methods, Bk is updated by theQuasi-Newton methods.

A local method, that can be practically implemented on the generallocal functions, is not presented. In this paper, we use the steepest descentdirection to construct the local model. The steepest descent direction forthe locally Lipschitz functions is an element of the Goldstein subgradi-ent with minimal norm. Based on the method, that approximate thisdirection, several bundle algorithms were developed [1-6]. The efficiencyof these algorithms depends on the approximation accuracy. To improvethe efficiency of an algorithm, a larger number of subgradients must becomputed to approximate the Goldstein subgradient efficiency and, thisis time consuming. For example, in [6], the steepest descent direction isapproximated by sampling gradients. This approximation is appropriate,but computing this approximation for large scale problems is very expen-sive. In [4], the steepest descent direction is iteratively approximated.This method computes a good approximation for the steepest descent di-rection by the less number of subgradients. The numerical results showedthat this algorithm is more efficient than other bundle methods.

By an approximation of the steepest descent direction, we proposean quadratic model for the locally Lipschitz functions. We combine theCauchy point and CG-Steihaug methods [7] to approximate the quadraticmodel solution. The numerical results show that the TR algorithm hasbetter behavior by this combination. In this paper, we implement thisalgorithm by Matlab code and compare its efficiency by other methods.

The nonsmooth trust region algorithm and its

convergence

In [8], the local model for locally Lipschitz functions is given as follow

m(x, p) = f(x) + φ(x, p) +1

2pTBp. (2)

26

Based on some assumption on φ(., .), the global convergent of TR wasproved. The authors purposed the following function

φ(x, p) = maxv∈∂f(x)

< v, p > .

But by this definition, minimization of the local model is impractical. Inthis paper, we give another local model for the locally Lipschitz functions.To construct the local model for the locally Lipschitz functions, we try tosubstitute the gradient in (1) by a suitable element of ∂ǫf(x).

Let ǫ > 0, the steepest descent direction is computed by using ∂ǫf(x).Consider the following function

v0 = arg minv∈∂ǫf(x)

‖v‖, (3)

and let d0 = − v0‖v0‖ . By Lebourg’s Mean Value Theorem, there exists

ξ ∈ ∂ǫf(x) such that

f(x+ d0)− f(x) = ǫξTd0 ≤ −ǫvT0v0‖v0‖

= −ǫ‖v0‖.

In fact, d0 is the steepest descent direction. But solving (3) is impractical,thus ∂ǫf(x) is approximated by its finite subset, i.e., if W ⊂ ∂ǫf(x) thenconvW is considered an approximation of ∂ǫf(x). Consider the followingproblem

vw = arg minv∈convW

‖v‖,

let d = − vw‖vw‖ . If f(x+ ǫd)− f(x) ≤ −cǫ‖vw‖ for some c ∈ (0, 1), then d

can be an approximation of a steepest descent direction. Else by addinga new element of ∂ǫf(x) in W , the approximation of ∂ǫf(x) is improved.The method, how construct such a subset, is described in [4].

Suppose that Wk ⊆ ∂ǫf(xk) and conv Wk is an approximation of∂ǫf(xk). We consider the following problem

‖vk‖ = arg minv∈conv Wk

‖v‖,

and suppose that f(xk − ǫ vk‖vk‖ )− f(x) ≤ −cǫ‖vk‖ where c ∈ (0, 1). In [4],

an algorithm is presented for finding Wk and vk. Based on this subdiffer-ential, vk ∈ ∂ǫf(xk), we define the following quadratic model:

m(xk, p) = f(xk) + vTk p+1

2pTBkp,

27

where Bk is a positive definite matrix. Based on this quadratic model,the trust region method is presented as follows.

Algorithm 1. (The nonsmooth trust region algorithm)

Step 0: Let ∆0,∆1 > 0, θ∆, δ1, θδ ∈ (0, 1), x1 ∈ Rn, ξ1 ∈ ∂f(x),

c1, c2, c3 ∈ (0, 1), c4 > 1, B1 = I and, k = 1.

Step 1: Apply Algorithm 2 in [4] at point xk with parameters ǫ = ∆k,δ = δk and c = c1. Suppose Algorithm 2 in [4] finds a properapproximation of ∂ǫf(xk), convWk, and a adequate subgradient,vk, such that

vk = arg minv∈convWk

‖v‖.

Step 2: If ‖vk‖ = 0, then stop, else if ‖vk‖ ≤ δk, then set ∆k+1 =θ∆ ×∆k, δk+1 = δk × θδ, xk+1 = xk, k = k + 1 and go to Step 1.Else set δk+1 = δk and go to Step 3.

Step 3: Solve the following quadratic subproblem:

minp∈Rn

m(xk, p) = f(xk) + vTk p+1

2pTBkp s.t. ‖p‖ ≤ ∆k,

and set pk be its solution.

Step 4: If f(xk + pk)− f(xk) ≤ c1vTk pk, then set xk+1 = xk + pk and goto Step 5, else set ∆k+1 = θ∆×∆k, xk+1 = xk, k = k+1 and go toStep 1.

Step 5: Define the following ratio

ρk =f(xk + pk)− f(xk)Q(pk)−Q(0)

.

If ρk ≥ c3 and ‖pk‖ = ∆k then, set ∆k+1 = min∆0, c4 ×∆k and,if ρ ≤ c2 then, set ∆k+1 = ∆k × θ∆. Else set ∆k+1 = ∆k.

Step 6: Select a subgradient ξk+1 ∈ ∂f(xk+1), then update Bk by theBFGS method. Set k = k + 1 and go to Step 1.

The following theorem proves the convergent of algorithm.

28

Theorem 1. Let f : Rn → R be a locally Lipschitz function. If thelevel set

L := x : f(x) ≤ f(x1)is bounded, then either Algorithm 1 terminates finitely at some k0 with‖vk0‖ = 0, or the sequence xk, generated by Algorithm 1, is convergent.If x∗ = limk→∞ xk, then 0 ∈ ∂f(x∗).

References

1. A. A. Goldstein. ”Optimization of Lipschitz continuous functions,” Mathe-matical Programming , 13:14–22, (1977).

2. D. P. Bertsekas and S. K. Mitter. A descent numerical method for opti-mization problems with nondifferentiable cost functionals,” SIAM Journal onControl, 11:637–652, (1973).

3. M. Gaudioso and M. F. Monaco. ” A bundle type approach to the uncon-strained minimization of convex nonsmooth functions,” Mathematical Pro-gramming, 23(2):216–226, (1982).

4. N. Mahdavi-Amiri and R. Yousefpour. ” An effective nonsmooth optimizationalgorithm for locally lipschitz functions,” Accepted Journal of OptimizationTheory Application.

5. P. Wolfe. ” A method of conjugate subgradients for minimizing non-differentiable functions,” Nondifferentiable Optimization, M. Balinski and P.Wolfe, eds., Mathematical Programming Study, North- Holland, Amsterdam,3:145–173, (1975).

6. J. V. Burke, A. S. Lewis, and M. L. Overton. ”A robust gradient samplingalgorithm for nonsmooth, nonconvex optimization,” SIAM Journal of Opti-mization, 15:571–779, (2005).

7. J. Nocedal and S. J. Wright Numerical optimization, Springer, (1999).

8. L. Qi and J. Sun. ” A trust region algorithm for minimization of locallylipschitzian functions,” Mathematical Programming , 66:25–43, (1994).

29

A Conjugate Gradient Projection Algorithm forsystems of Large-Scale Nonlinear Monotone

Equations

Keyvan Amini1, Masoud Ahookhosh2, Somayeh Bahrami3

1 Department of Mathematics, Razi University, Kermanshah, Iran;

[email protected] Department of Mathematics, Razi University, Kermanshah, Iran;

[email protected] Department of Mathematics, Razi University, Kermanshah, Iran;

[email protected]

Abstract

Systems of nonlinear equations generally are a family of prob-lems that is so close to optimization problems and often arise in theapplied sciences, technology and industry. In general, the systemof nonlinear equations can be formulated mathematically by

G(x) = 0, subject to x ∈ Rn, (1)

where G : Rn → Rn is a continuous function. In particular, thenonlinear monotone equations are a class of nonlinear equationswhenever G(x) satisfies the following monotonicity condition

(G(x)−G(y))T (x− y) ≥ 0, for all x, y ∈ Rn,

guaranteeing that the solution set of (1) is a convex set.We propose two derivative-free approaches for solving a large-scalenonlinear monotone system. The framework firstly generates a spe-cific direction then employs a line search to construct a new point.If the new point doesn’t solve the problem, the projection techniqueconstructs an appropriate hyperplane that separates the current it-erate from the solutions of the problem. Then the projection ofthe new point onto the hyperplane will determine the next iterate.Thanks to the low memory requirement, we use two new conjugategradient directions. The global convergence is established Underappropriate conditions.

30

Software implementation of an algorithm for findingthe optimal control using a graphics accelerator

Anton Anikin1

1 ISDCT SB RAS, Irkutsk, Russia; [email protected]

An optimal control problem with the system, linear in the control

x = f1(x, t) + f2(x, t) · u (1)

and parallelepiped restrictions on the control action is considered. Need tofind minimum of terminal functional ϕ(x(t1)), where [t0, t1] - the time in-terval of the process. In this formulation, with maintaining the regularityof the optimality conditions, controls which satisfies the Pontryagin max-imum principle, have the relay character. The report proposes a methodof finding the global extremum of the terminal objective function, basedon the calculation of the switching points of optimal control.

Parameterization of the scalar control action that allows to construct avalid control with predetermined number of switching points is proposed.The first parametrization parameter includes not only the value of the firstswitching point, but also points to the border - the bottom or top, fromwhich the constructed control begins. Because of this first parametrizationparameter changes on then interval [t0, 2 · t1], the rest parameters - on theinterval [t0, t1]. An example of such parametrization approach for casewith 2 switching points presented on Fig. 1.

Fig. 1

Fig. 2 presents constructed relay and piecewise linear controls fromsome selected switching points. An algorithm for solving the optimal

31

Fig. 2

control problem consists of a sequence of nonconvex unconstrained mini-mization problems with an increasing number of variables correspondingto the desired number of switching points. The optimal solutions of theauxiliary finite-dimensional problems make up a monotonic sequence ofvalues converging to the minimum value of functional in problems withfinite number of switching points in optimal control.

The search algorithm based on the sequential solution of nonconvexproblems of one-dimensional search on a random direction is proposedto solve the unconstrained minimization problem on the hypercube. Foreach random direction is calculated interval of variation that would guar-antee to find any solution within the allowable box and the problem isformulated as a one-dimensional search which is non-convex in the gen-eral case. Search for global minimum in the direction is performed usingan algorithm based on a combination of spline-search proposed in [1], andreliable, but slow Strongin classical method [2]. An example of such globalsearch for 2-point controls is presented on Fig. 3.

Parallelization of the algorithm performed with using CUDA technol-ogy [3] by accelerating multiple function calculation during on-dimensionalsearches. Such calculations require solutions of the Cauchy problem. Nu-merical experiments performed on Nvidia GPUs (Tesla and Fermi gener-ation) with using a single (float) and double precision confirms the highpotential of parallelism of the algorithm, results for some test problemspresented in Table 1.

The authors were supported by the Russian Foundation for Basic Research

(project no. 10-01-00595).

32

Fig. 3

Table 1: Numerical experiment results; 1024 · 1024 intergation

Problem GPU (s) CPU (s) CPU / GPU1 8.6 149.0 17.32 21.0 383.1 18.23 14.6 241.5 16.5

References

1. A.Yu. Gornov. “Using spline-approximation to design optimization algo-rithms with new computational properties,” Proceedings of the all-Russiaconference “Discrete optimization and operations research”, Vladivostok. p.99 (2007).

2. R.G. Strongin Numerical Methods for Multiple-optimization, Nauka, Moscow(1978).

3. J. Sanders, E. Kandrot CUDA by Example: An Introduction to General-Purpose GPU Programming, Addison-Wesley, Boston (2011).

33

The reduction of the optimal parametric synthesis tothe linear programming problem

Maxim Anop1, Yaroslava Katueva2

1 Institute of Automation and Control Processes of FEB RAS, Vladivostok,

Russia; [email protected] Institute of Automation and Control Processes of FEB RAS, Vladivostok,

Russia; [email protected]

The engineering system parameters are subject to random variationsand the variations may be considered as non-stationary stochastic pro-cesses. The conventional methods for choosing parameters (parametricsynthesis) generally do not take account of parameters field deviationsfrom their design values. As a result the engineering systems designedin such a manner are not optimal in the sense of their gradual failurereliability.

Suppose that we have a system which depends on a set of n inputparameters x = (x1, . . . , xn)

T . The structure of the system determinesthe dependence of the output parameters of the internal parameters y(x).It is considered that the equations y(x) are described with a model thatis given in any form such as analytical equations, algorithmic form orsimulation model.

We will say that system is acceptable if y(x) satisfy the conditions (1):

a ≤ y(x) ≤ b, (1)

where y, a and b are m-vectors of system responses (output parameters)and their specifications, e.g. y1(x) – average power, y2(x) –delay, y3(x) –gain.

The inequalities (1) define a region Dx in the space of input (system)parameters

Dx = x ∈ Rn|a ≤ y(x) ≤ b (2)

Dx is called the performance region for the system.The engineering system parameters are subject to random variations

(aging, wear, temperature) and the variations may be considered as stochas-tic processes:

X(t) = X1(t), . . . , Xn(t). (3)

34

In general parametric reliability optimization problem (optimal para-metric synthesis) can be stated as follows [1].

The equations y(x), conditions of acceptability (1) and a service timeT are given. The task is to find such a deterministic vector of parame-ter ratings (nominal values) xnom = (x1nom

, x2nom, . . . , xnnom

)T that thereliability

xnom = argmaxPX(xnom, t) ∈ Dx, ∀t ∈ [0, T ]) (4)

are maximal.The practical algorithm of the stochastic criterion calculation is based

on the conventional Monte-Carlo method [1]. In fact the distribution lowsof system parameters variations and the characteristics of random param-eters degradation processes X(t) are often unknown. The replacementof original stochastic criterion with a certain deterministic one is usedin case of uncertainty conditions allows nearby optimum solutions to beobtained. It is a so-called a “minimal serviceability reserve”, the largestdistance from the region margins and e.t.c. [1-2].

The region of acceptability, and its border are not analytically givengenerally. In this case, the problem (4) can be formulated as follows:

xnom = argmax(d(xnom, ∂Dx),xnom ∈ Dx) (5)

where d(x, ∂Dx) is a distance from x to the boundary ∂Dx measured withany way. The solution of the problem (5) is the center of the inscribed inthe region Dx figure with maximum norm.

The modification of the simplicial approximation method offered bythe S.W. Director and G.D. Hetchell [3] is discussing in the paper.

The first step in parametric synthesis problem is narrowing the searcharea in a space of internal parameters. A circumscribed parallelepipedis constructed for this purpose. Proposed in [4] the algorithm based onMonte-Carlo method enables to receive the points of contact with minimalK−i and maximalK+

i coordinates for each coordinate direction, belongingboth to circumscribed box and region of acceptability.

The second step is the construction of the piecewise linear internal ap-proximation Dx for region of acceptability Dx. Let’s assume that pointsp1, . . . , pN belonging to the border of ∂Dx are received. Then the convexhull of this set pj ∈ ∂Dx, j = 1, 2, . . . , N will give required approxima-tion Dx. It is possible to use points of a contact K−

i ,K+i , i = 1, 2, . . . , 2n

as the set pj ∈ ∂Dx, j = 1, . . . , N .

35

The convex polytope of contact’s point enables to reduce the prob-lem (5) to a linear programming problem. The maximum figure (cube,ellipsoid) will be solution of this task. The center of it will be a requiredvector of nominal parameters.

The authors were supported by FEB RAS: Grants 12-I-OEMMPU -01 (Basic

Research Program of PMMCP Branch RAS no.14).

References

1. O.V.Abramov and K.S.Katuyev Effective methods for parametric synthesis ofstochastic systems,First Asian Control Conference. 3: 587-589, 1994.

2. Abramov O.V., Katueva Y.V. and Nazarov D.A.Construction of acceptabilityregion for parametric reliability optimization. Reliability & Risk Analysis:Theory & Applications. 10: 20-28, 2008.

3. S.W. Director and G.D. Hachtel “The simplicial approximation approach todesign centering and tolerance assignment”, IEEE Trans. Circuits Syst., vol.CAS-24, pp. 363-371, 1977.

4. M.Anop, Y.Katueva GEOMETRIC ANALYSIS OF PERFORMANCE RE-GION BASED ON THE MONTE-CARLO METHOD // Proceedings of the7th International Conference on Mathematical Methods in Reliability; The-ory. Methods. Applications (MMR2011), edited by Lirong Cui&Zian Zhao.Beijing: Beijing Institute of Technology Press. 2011, pp.244-247.

Boundary value games in optimal control

Anatoly Antipin1

1 Computing Center of Russian Academy of Sciences, Moscow, Russia;

[email protected]

1. Statement of problem in finite dimensional space.Consider a two-person game, which is the problem of computing a

fixed point x∗0 = (x∗10, x∗20) of extreme inclusions

x∗10 ∈ Argminf10(x10, x∗20) + ϕ1(x10) | C10x10 ≤ c10, x10 ∈ X10,

x∗20 ∈ Argminf20(x∗10, x20) + ϕ2(x20) | C20x20 ≤ c20, x20 ∈ X20, (1)

36

where X10 ⊂ Rn11 and X20 ⊂ Rn2

2 are closed convex sets in finite dimen-sional Euclidean spaces. The objective functions f10(x10, x20) + ϕ1(x10)and f20(x10, x20)+ϕ2(x20) are defined on the Cartesian product of spacesRn1

1 and Rn22 . All functions are convex in own variables, i.e. first func-

tion is convex in the variable x10, the second – in the variable x20 for allx10 ∈ X10 and x20 ∈ X20. For the first player x20 ∈ X20 is the parameter,for the second player x10 ∈ X10, on the contrary, is the parameter. Ifboth sets are compact, then there is always a solution x∗0 = (x∗10, x

∗20) of

the game (1).The meaning of the solution of this game lies in the fact that none of

the players are not interested in breach of its state otherwise the value ofits objective function can only increase. It seems convenient to scalarizethe problem (1) and instead of the system of parametric optimizationproblems compute a fixed point of the extremal mapping.

For this purpose, we introduce a normalized function of the form

Φ0(v0, w0)+ϕ0(w0) = f10(z10, x20)+ϕ1(z10)+f20(x10, z20)+ϕ2(z20), (2)

where w0 = (z10, z20), v0 = (x10, x20), v0, w0 ∈ W0 = X10 ×X20. In thenew variables, two-person game with a Nash equilibrium is transformedinto the problem of computing the fixed points of extremal mapping

v∗0 ∈ ArgminΦ0(v∗0 , w0) + ϕ0(w0) | w0 ∈W0. (3)

By the separability of the function Φ(v, w) with respect to the variablesw0 = (z10, z02) solution of problem (3) is a solution of problem (1), butnot vice versa.

If the neighborhood of the fixed point in problem (3) has a saddlestructure, the saddle-point methods such as extraproximal or extragradi-ent methods converge to the solution of this problem.

Let us consider the differential analogue of the problem (1) in func-tional space. This game is considered on a fixed time interval [t0, t1] withfree ends, and linear differential systems. On the sets of attainabilitygenerated by free right ends (x1(t1) = x11, x2(t1) = x21) of the trajecto-ries (x1[u1(t)], x2[u2(t)]) = (x1(t), x2(t)), the payoff functions are defined,u1(t), u2(t) ∈ U1×U2 ⊂ PC[t0, t1] are set of piecewise continuous controls,x1(t), x2(t) ∈ X1×X2 ⊂ PC[t0, t1]1 are set of piecewise continuously dif-ferentiable trajectories. A formal statement of the problem has the form:

37

the first player

d

dtx1(t) = D1(t)x1(t) +B1(t)u1(t), x∗10 ∈ X1(t0), (4)

U1 = u1(t) ∈ Lr2[t0, t1]| u1(t) ∈ [u−1 , u+1 ], t0 ≤ t ≤ t1, (5)

x∗11 ∈ Argminf1(x11, x∗21) + ϕ1(x11) | C11x11 ≤ c11, x11 ∈ X1(t1), (6)

the second player

d

dtx2(t) = D2(t)x2(t) +B2(t)u2(t), x∗20 ∈ X2(t0), (7)

U2 = u2(t) ∈ Lr2[t0, t1]| u2(t) ∈ [u−2 , u+2 ], t0 ≤ t ≤ t1, (8)

x∗21 ∈ Argminf2(x∗11, x21) + ϕ2(x21) | C21x21 ≤ c21, x21 ∈ X2(t1), (9)

whereX1(t1) ⊂ Rn11 , X2(t1) ⊂ Rn2

2 . Here x∗10, x∗20 stand for the initial con-

ditions, which form two-person game (1) solution. The pair x∗11, x∗21 is a

solution of the terminal two-person game (6),(9) with a Nash equilibrium.The dynamics (4),(5) and (7),(8) takes the system (4)–(9) from the

initial state to the terminal one.Both games are relatively independent: the interests of the players are

connected only by payoff functions, but not by their dynamics. Note thatthe payoff functions describe the overall interest of each player: ϕ1(x11),ϕ2(x21) – interests, where the players are not going to make concessions,f1(x11, x21), f2(x11, x21) – interests, where the players are willing to makeconcessions to find a compromise.

2. The problem of calculating the fixed points of extremalmapping. Let us imagine a game of two persons (4)–(9) in an ag-gregated form. For this purpose we introduce the new macro variablesw(t) = (x1(t), x2(t))

T , u(t) = (u1(t), u2(t))T and represent the controlled

dynamics on the Cartesian product X1(t)×X2(t) in the form

(x1(t)x2(t)

)=

(D1(t) 00 D2(t)

)(x1(t)x2(t)

)+

(B1(t) 00 B2(t)

)(u1(t)u2(t)

),

w(t) =

(x1(t)x2(t)

)∈(X1(t)X2(t)

)=W (t), u(t) =

(u1(t)u2(t)

)∈(U1

U2

)= U.

38

Then we introduce the aggregate terminal variables w1 = (z11, z21)T , v∗1 =

(x∗11, x∗21)

T and the payoff function

Φ1(v∗1 , w1) + ϕ1(w1) = f11(z11, x

∗21) + f21(x

∗11, z21) + ϕ1(z11) + ϕ2(z21),

and represent the aggregate terminal problem in the form

Φ1(v∗1 , v

∗1) + ϕ1(v

∗1) ≤ Φ1(v

∗1 , w1) + ϕ1(w1),

C1 =

(C11 00 C21

)(x11x21

)≤(c11c21

);

(x11x21

)∈(X1(t1)X2(t1)

)=W1.

Now, the problem of calculating the fixed point v∗1 ∈ W1 of extremalmappings has the form

d

dtw(t) = D(t)w(t) +B(t)u(t), v(t0)) = v∗0 , u(t) ∈ U, t0 ≤ t ≤ t1, (10)

Φ1(v∗1 , v

∗1) + ϕ1(v

∗1) ≤ Φ1(v

∗1 , w1) + ϕ1(w1),

C1w1 ≤ c1, w1 ∈ W1 ⊂ R2n. (11)

The extremal mapping can be written in explicit form, then

d

dtw(t) = D(t)w(t) +B(t)u(t), w(t0) = v∗0 , u(t) ∈ U,

v∗1 ∈ ArgminΦ1(v∗1 , w1)+ϕ1(w1) | C1w1 ≤ c1, w1 = w(t1) ∈ W1 ⊂ R2n,

(12)Systems (10),(11) or (12) are aggregated form of the game (4)–(9). Fora fixed parameter v1 = v∗1 the resulting system is a convex program-ming problem, formulated in a functional space with respect to finite-dimensional w1 = w(t1) ∈W1 and functional w(t), u(t) variables.

In the regular case, the Lagrange function

L1(v∗1 , p1, w1, ψ(t), w(t), u(t)) = Φ1(v

∗1 , w1) + ϕ1(w1) + 〈p1, C1w1 − c1〉+

+

∫ t1

t0

〈ψ(t), D(t)w(t) +B(t)u(t) − d

dtw(t)〉dt

defined for all p1 ≥ 0, w1 ∈W1, ψ(t) ∈ PC[t0, t1]′

, w(t) ∈ PC1[t0, t1], u(t) ∈U ⊂ PC[t0, t1], where p1 ψ(t) are dual variables, and (w(t), u(t)) are pri-mal variables, has a saddle point (p∗1, ψ

∗(·)), (v∗1 , v∗(·), u∗(·)). This pointsatisfies the system of inequalities

Φ1(v∗1 , v

∗1) + ϕ1(v

∗1) + 〈p1, C1v

∗1 − c1〉+

39

+

∫ t1

t0

〈ψ(·), D(t)v∗(·) +B(t)u∗(·)− d

dtv∗(·)〉dt ≤

≤ Φ1(v∗1 , v

∗1) + ϕ1(v

∗1) + 〈p∗1, C1v

∗1 − c1〉+

+

∫ t1

t0

〈ψ∗(·), D(t)v∗(·) +B(t)u∗(·)− d

dtv∗(·)〉dt ≤

≤ Φ1(v∗1 , w1) + ϕ1(w1) + 〈p∗1, C1w1 − c1〉+

+

∫ t1

t0

〈ψ∗(·), D(t)w(·) +B(t)u(·)− d

dtw(·)〉dt

for all p1 ∈ Rn+, ψ(·) ∈ PC1[t0, t1]′

, v1, w(·) ∈ PC1[t0, t1], u(·) ∈ U ⊂PC[t0, t1].

From the resulting system of saddle-point inequalities by using rela-tively simple arguments, we can get the dual problem in form:

d

dtψ∗(t) +DT (t)ψ∗(t) = 0, ψ∗

1 = ∇2Φ(v∗1 , v

∗1) +∇ϕ1(v

∗1) + CT1 p

∗1,

∫ t1

t0

〈BT (t)ψ∗(t), u(t)− u∗(t)〉dt ≥ 0, u(t) ∈ U.

3. The boundary value problem and the method for its solu-tion.

Combining the primal and dual problems, we obtain the boundaryvalue problem

d

dtv∗(t) = D(t)v∗(t) +B(t)u∗(t), v(t0) = v∗0 ,

〈p1 − p∗1, C1w∗1 − c1〉 ≤ 0, p1 ≥ 0,

d

dtψ∗(t) +DT (t)ψ∗(t) = 0, ψ∗

1 = ∇1Φ(v∗1 , v

∗1) +∇ϕ1(v

∗1) + CT1 p

∗1,

∫ t1

t0

〈BT (t)ψ∗(t), u(t)− u∗(t)〉dt ≥ 0, u(t) ∈ U. (13)

To solve this system, including differential equations and variationalinequalities, we use the saddle-point method in the form of an extragradi-ent process. This method can be treated as a controlled method of simpleiteration. In this method, each iteration consists of two half-steps. The

40

first half-step can be interpreted as a control in the form of feedback. Theformulas of this iterative process have the form:

1) prediction half-step

d

dtvn(t) = D(t)vn(t) +B(t)un(t), vn(t0) = v∗0 ,

pn1 = π+(pn1 + α(C1v

n1 − c1)),

d

dtψn(t) +DT (t)ψn(t) = 0, ψn1 = ∇1Phi(v

n1 , v

n1 ) +∇ϕ(vn1 ) + CT1 p

n1 ,

un(t) = πU (un(t)− αBT (t)ψn(t)); (14)

2) basic half-step

d

dtvn(t) = D(t)vn(t) +B(t)un(t), vn(t0) = v∗0 ,

pn+11 = π+(p

n1 + α(C1v

n1 − c1)),

d

dtψn(t) +DT (t)ψn(t) = 0, ψn1 = ∇1Φ(v

n1 , v

n1 ) +∇ϕ(vn1 ) + CT1 p

n1 ,

un+1(t) = πU (un(t)− αBT (t)ψn(t)). (15)

It follows from this process, that the differential equations are only usedfor the calculation of conjugate functions ψn(t), ψn(t). Therefore, theprocess can be written in a more compact form

pn1 = π+(pn1 + α(C1v

n1 − c1)),

un(t) = πU (un(t)− αBT (t)ψn(t)),

pn+11 = π+(p

n1 + α(C1v

n1 − c1)),

un+1(t) = πU (un(t)− αBT (t)ψn(t)), (16)

where ψn(t) and ψn(t) are computed as solutions of the differential sys-tems. It has been proven that the process converges monotonically in thenorm of controls space to one of the solutions of original problem.

Theorem. If the set of solutions (13) is not empty and belongs tothe subspace PC[t0, t1]×PC1[t0, t1], the functions Φi(v

∗i , wi)+ϕi(wi), i =

41

1, 2, are positive semidefinite, and convex in the variables wi, differen-tiable with respect to these variables, whose gradients satisfy the Lips-chitz conditions, then the sequence of approximations generated by theprocess (14),(15) with the choice of the parameter α from the condition0 < α < α0, decreases monotonically in the norm on the Cartesian prod-uct of variables (controls, trajectories and variables of terminal problems).At the same time, any weakly converging subsequence of controls uni(t)weakly converges to the optimal control u∗(t), and a corresponding subse-quence of trajectories vni(t) converges to optimal trajectory v∗(t) in theuniform norm Cn[t0, t1].

If the sequens of controls un(t) has a strong limit point in the normof Ln2 , then the process (vn(t), un(t)) converges to a solution (v∗(t), u∗(t))monotonically in the norm spaces Ln2 × Lr2.

In the method (14),(15) the vector v∗0 of initial conditions is used, andit must first be calculated by solving the equilibrium problem (1).

Minimization of maximum lateness for M stationswith tree topology

Dmitry Arkhipov1, Alexander Lazarev2, Elena Musatova3

3 Institute of Control Sciences, Moscow, Russia; [email protected] Institute of Control Sciences, Moscow, Russia; [email protected]

3 Institute of Control Sciences, Moscow, Russia; [email protected]

Minimization maximum weighted lateness for 2 stations. Thefollowing problem of scheduling theory is considered. There is a set oforders (wagons) N . Each order j ∈ N releases on the station A at themoment rj . Due date of order j equals dj = rj + δ. Each order hasits own value wj > 0. Wagons are delivered to the station B by train,which covers the distance between A and B in time p. Each train canbe departed after the time α of the previous departure. Our goal is totransport all wagons on the station B. The objective function is

minmaxj∈N

(wjLj) (1)

We also formulate the ancillary problem with the objective function:

minCmax|(wL)max < y (2)

42

Schedule which holds (2) we would call Θ(N, y).For each values j and y we can determine the moment t′j = rj+δ+

ywj−p.

Order j must depart from the station A in the moment which belongs tothe interval [rj , t

′j). We define a set of orders which must be transported

to the station B on the train which number is not exceeding m as Sm.Note that on the first step of algorithm sets S1, . . . , Sm−1 are empty andset Sm is full of orders 1, . . . , n.Property 1. We consider the train m which departs at the moment tm

in the schedule Θ(N, y).(i) If at the moment tm there are more than k orders then we shouldtransport on the train m k jobs with minimal t′j .(ii) Trainm can depart at the moment tm holds tm ≥ max(rkm, r(Sm), tm−1+α).(iii) All orders Jl which holds tm + α ≥ t′l must be transported on thetrains which numbers are not bigger than m, so Jl ∈ Sm.(iv) When the train m was departed, all orders from the set Sm must hadalready depart.Algorithm 1. On each step of algorithm we try to depart the train mfrom the station A. Firstly, we choose the moment tm which holds (ii).Secondly, we choose k orders which would be transported on train m withhelp of (i). After that we check if (iii) holds. If there exists an orderJl : rl > tm, tm + α ≥ t′l, so according to (iii) we have to include thisorder Jl into sets Sm, Sm+1, . . . , Sq, then let us return to checking (ii). Ifthere is no such order Jl, (iii) must hold, except the case when we havex > k released orders from the set Sm at the moment tm (this ordersare .... X0). We obtain x − k orders from X0, that have to be trans-ported on the trains with number lower than m. After looking for ordersJj , t

′j > tm+α in the sets Tm−1, Tm−2, . . . until we found x−k orders hold

this property (let the last one was founded in the train s). After we obtainthree sets of orders: Tm−1

s = Ts ∪ · · · ∪Tm−1, X′ - set of jobs which holds

j|t′j > tm + α, j ∈ Tm−1s with minimal x moments t′ from all such jobs

j, and a set X0. Let us consider the set X = (Tm−1s \X ′) ∪ X0. Orders

from this set must be transported on trains s, . . . ,m. When we departthis orders we have to change rki on r(X(i−s+1)k) in the property (ii)because we can depart only orders from X . Property (i) holds because allorders which are not belongs to X and released until this moment holdst′ > tm + α. Properties (iii) and (iv) hold, except the cases when one of

43

sets Si changes, if we face it we should go to the next step - consideringthe train i. If we don’t face problems during the transporting set X weshould go to the next step - considering the train m+ 1. This algorithmterminates if on any step we obtain the set Si with more than ki orders.Theorem 1. Algorithm 1 constructs the schedule Θ(N, y) according tocriterion Cmax|(wL)max < y. If algorithm 1 was terminated, there are noschedule π holds (wL)max < y.Lemma. If there are exist two schedules Θ(N, y1) and Θ(N, y2) (y1 > y2)constructed with help of the algorithm 1, then for each i = 1, . . . , qand a pair of sets Si(Θ(N, y1)) and Si(Θ(N, y2)) holds Si(Θ(N, y1)) ⊆Si(Θ(N, y2))Algorithm 2. Firstly we construct a schedule in which each train de-parts as soon as possible. then we consider the order j with maximalwjLj. Order j transports on the train m. If we want to improve theobjective function we must transport order j on the train which numberis lower than m, so j ∈ Sm−1. On the next step we construct the sched-ule Θ(N,wjLj).We should repeat this operation until we construct theschedule Θ(N, y′) with the objective function y0, when schedule Θ(N, y′)doesn’t exist. On this step we note that Θ(N, y′) is an optimal schedulewith the objective function y0.Theorem 2. Algorithm 2 constructs the schedule π which is optimal ac-cording to criterion (wL)max and has minimal Cmax among all scheduleswith the objective function (wL)max.Minimization maximum lateness for 3 stations. There are threestations A,B,C and three sets of orders NAB, NAC , NBC . The orderj ∈ NAC releases at the moment rACj on the station A and must betransported to the station C. The due date of this order we define asdACj = rACj + δAB + δBC . Parameters of other orders defines similarly.

Train covers the distance between A and B in time pAB and the distancebetween B and C in time pBC . There are k wagons in each train. Eachtrain can be departed after time α of the previous departure. Let ussuppose that δAB > pAB + α and δBC > pBC . the objective function is

min( maxj∈NAB∪BC∪AC

(Lj)) (3)

We also formulate the ancillary problem with the objective function

min(Cmax)|Lmax < y (4)

44

The schedule which holds (4) we would call Θ3(N, y), N = NAB ∪NAC ∪NBC .Algorithm 3. Firstly, we construct intervals for each order and eachtrack (AB and BC). The order j ∈ NAB must be transported on B be-fore the moment rABj +δAB+y, so we obtain that it’s interval on the track

AB is [rABj , rABj + δAB+ y−pAB). Order j ∈ NBC , corresponds with the

interval [rBCj , rBCj +δBC+y−pBC) on the track BC. The order j ∈ NAC

corresponds with the interval [rACj , rACj + δAB + δBC + y − pAB − pBC)on the track AB, because this order must be departed from B beforethe moment rACj + pAB. We also obtain that order j corresponds with

the interval [rACj + pAB, rACj + δAB + δBC + y − pBC) on the track BC.Each job must start it’s transportation on the track in time belongs to theinterval which is corresponds with this track. When all interval are con-structed we use the algorithm 1 to construct the schedule Θ(NAB∪AC , y)corresponds with the track AB and the schedule Θ(NBC∪AC , y) corre-sponds with the track BC, subject to constructed intervals. If one ofthis schedules isn’t constructed successfully our algorithm terminates. Ifnot we pay attention for ”bad” orders j which completion time Cj in theschedule Θ(NAB∪AC , y) more than it’s time of departure in the scheduleΘ(NBC∪AC , y). If there are no ”bad” orders we should depart orders oneach track according to it’s own schedule Θ. If there are exist some ”bad”orders we should find the first of them. We can use two methods to getout of the ”bad” order.Method 1.We don’t change moments of departure of trains on the trackAB.

45

Method 2.We change the moments of departure of trains on the trackAB.

We use the following scheme to choose the right method.

If there are no ”bad” orders in our pair of schedules Θ(NAB∪AC , y) andΘ(NBC∪AC , y), then we depart trains on the track AB according toΘ(NAB∪AC , y) and on the track BC according to Θ(NBC∪AC , y). Asa result we obtain the schedule Θ3(N, y).Theorem 3. Algorithm 3 constructs the schedule Θ3(N, y). If algorithm3 terminates then there is no schedule which holds (4).To construct the optimal schedule we use algorithm 2. The only differenceis that we should use the schedule Θ3(N, y) instead of Θ(N, y).M station with tree topology. The formulation of this problem andthe problem for 3 stations is the same. The only difference is that wedeals with M stations with tree topology. Due to the tree topology thereis only one way between each pair of stations. We also can enumeratestations from left to right (or from right to left).Algorithm 4.We use algorithm 3 to get out of ”bad” orders for each

46

station, from the left to the right follows the numeration (according tochosen direction of the train moving). When there are no ”bad” orderson each station we obtain the schedule ΘM (N, y). After that we use al-gorithm 2 to construct the optimal schedule for M stations.Theorem 4. Algorithm 4 constructs the optimal schedule according to

criterion Lmax in O(M2 n4

k ) operations.


(project no. 11-08-13121).

Existence theorems for elliptic equations inunbounded domains

Armen Beklaryan1

1 Lomonosov Moscow State University, Moscow, Russia;

[email protected]

We consider the first boundary value problem for elliptic systemsdefined in unbounded domains, which solutions satisfy the condition offiniteness of the Dirichlet integral also called the energy integral

∫

Ω

|∇u|2dx <∞.

Basic concepts

Let Ω is an arbitrary open set in Rn. As is usual, by W 1

2, loc(Ω) we denotethe space of functions which are locally Sobolev, i.e.

W 12, loc(Ω) = f : f ∈ W 1

2 (Ω ∩Bxρ ), ∀ ρ > 0 , ∀x ∈ Rn ,

where Bxρ – open ball with center at point x and with radius ρ. If x = 0

then we will write Bρ. We will denote byo

W12, loc(Ω) set of functions from

W 12, loc(R

n), which is the closure of C∞0 (Ω) in the system of seminorms

47

‖u‖W 12 (K), where K ⊂ R

n are various compacts. Let denote by L12(Ω)

a space of generalized functions in Ω, which first derivatives belong toL2(Ω) [4], in other words

L12(Ω) = f ∈ D

′

(Ω) :

∫

Ω

|∇f | 2dx <∞.

Let ω ⊆ Rn is an open set, K ⊂ ω is a compact. We will denote

by Φϕ(K, ω) the set of functions ψ ∈ C∞0 (ω) such that ψ = ϕ in the

neighborhood of K, or in other words ψ − ϕ ∈o

W12, loc(R

n \ K).Let’s define a capacitance of a compact K relative to the set ω [4]:

capϕ(K, ω) = infψ∈Φϕ(K, ω)

∫

ω

|∇ψ| 2dx .

The capacitance of arbitrary closed set E ⊂ ω in Rn is defined by the

formula capϕ(E,ω) = supK⊂E

capϕ(K, ω). If ω = Rn, then instead of

capϕ(E,Rn) we will write capϕ(E).

Problem statement

Let L is a divergent operator

L =

n∑

i,j=1

∂

∂xi

(aij(x)

∂

∂xj

),

where aij are bounded measurable functions in Rn satisfying condition

γ|ξ|2 ≤n∑

i,j=1

aij(x) ξi ξj , ξ ∈ Rn, γ > 0 .

The solution of the Dirichlet problemLu = 0 in Ω

u|∂Ω = ϕ,(1)

where ϕ ∈W 12, loc(R

n), is a function u ∈ W 12, loc(Ω) such that:

48

1) u − ϕ ∈o

W12, loc(Ω), i.e. (u − ϕ)µ ∈

o

W12(Ω) for any function

µ ∈ C∞0 (Rn);

2) function u has bounded Dirichlet integral∫

Ω

|∇u|2dx <∞ ;

3) ∫

Ω

n∑

i,j=1

aij(x)∂u

∂xj

∂ψ

∂xidx = 0

for any function ψ ∈ C∞0 (Ω).

Basic results

Theorem 1. Let’s capϕ− c(Rn \Ω) <∞ for some constant c ∈ R

n. Thenthe problem (1) has a solution.Theorem 2. Let the problem (1) has a solution and it is true that

∫

Rn\Ω

|∇ϕ|2dx <∞ .

Then there is such constant c ∈ Rn, that capϕ− c(R

n \ Ω) <∞.Theorem 3. Let n ≥ 3. Then capϕ− c(R

n \ Ω) <∞ if and only if

∞∑

k=N

capϕ− c((B2k+1 \B2k−1) ∩ (Rn \ Ω), B2k+2 \B2k−2) <∞

for some N ∈ N.

Particular cases

Let consider the space Rn with a set of coordinates (x1, x2, . . . , xn) and

let ϕα = (1+|x1|)α. Domain Ω1,i is upper half-plane relative to xi, wherei 6= 1, in other words Ω1,i = (x1, x2, . . . , xn)|xi ≥ 0, i 6= 1. Domain Ω2

is the outer part of the space formed by surface of revolution relative tox1 of the curve from Fig.1.

49

x2 = |x1|β, β < 0

x1A

x2

Fig. 1: Domain Ω2

Corollary 1. Let n ≥ 2. Then for the domain Ω1,i and for boundedfunction ϕα the existence of solutions of the problem (1) is equivalent to

either an inequality α < −1

2or α = 0.

Corollary 2. Let n ≥ 3. Then for the domain Ω2 and for boundedfunction ϕα the existence of solutions of the problem (1) is equivalent to

either an inequality α < −1 + β(n− 3)

2or α = 0.

References

1. A.L. Beklaryan. “The first boundary value problem for the Laplace equationin unbounded domains,” Abstracts of the OPTIMA-2011, 2011.

2. A.A. Kon’kov. “The dimension of the space of solutions of elliptic systems inunbounded domains,” Journal, Mat. sbornik, V.184, No.12, 23–52, 1993.

3. O.A. Ladyzhenskaya, N.N. Ural’tseva. Linear and quasilinear elliptic equa-tions, M.: Nauka, 1964.

4. V.G. Maz’ya. Sobolev spaces, L.: Izdat. Leningr. Univer., 1985.

50

On the determination of the earthquake slipdistribution via linear programming techniques

Vladimir Bushenkov1, Bento Caldeira2, Georgi Smirnov3

1 University of Evora, Evora, Portugal; [email protected] University of Evora, Evora, Portugal; [email protected]

3 University of Minho, Braga, Portugal; [email protected]

The description that one can have of the seismic source is the mani-festation of an imagined model, obviously outlined from Physic Theoriesand supported by mathematical methods. In that context, the modellingof earthquake rupture consists in finding values of the parameters of theselected physics-mathematical model, through which it becomes possibleto reproduce numerically the records of earthquake effects on the Earthssurface. Actually, these effects are the elastic records at near field sourceand at far field source, and inelastic deformations recorded by geodetictechniques. The detail and accuracy level, with which the characteristicparameters for large earthquakes are computed, depends on the combina-tion of two factors - the applied methods and the used data.

Under the hypothesis of constant slip direction and constant rise timeof individual source time function, the problem of complete seismic sliptime history and distribution reconstruction reduces to the solution of asystem of linear equations. It is well-known that this inverse problemis ill-posed [6]. The usual regularization techniques [8] can hardly beapplied in this case because of a very high dimension of this problem (see,e.g., [3]). The problem can be overcome by introducing some additionalregularizing constraints. Some additional physical hypotheses, like no-backslip constraint, result in condition of non-negativeness of solutions tothe system of linear equations.

The positivity that prohibits negative seismic moment values, is aconstraint naturally assumed when used the Non Negative Least Squaresalgorithm (NNLS) [5] to inverts seismic waveforms to slip distribution(e.g., [7]).

We present and test a Linear Programming (LP) inversion in dualform, for reconstructing the kinematics of the rupture of large earthquakesthrough space-time seismic slip distribution on finite faults planes. Theproposed method can be considered as a continuation of the work started

51

in [2]. The proposed algorithm uses strong ground motion waveforms, butit can also used with other types of data as teleseismic waveforms as wellas with geodesic data (static deformation). We test the method with dataobtained by application to a synthetic model of rupture. To compare it,we rehearsed reconstructions with same data, but made by other stronglyused algorithms. Green functions (see, e.g., [1]) were calculated by a finitedifferences method applied to a 3D structure model [4].

The hypothesis of constant slip direction in general is not verifiedand the ”real” seismic slip time history and distribution reconstructionbecomes an hard nonlinear problem. In this work we suggest an algorithmfor seismic slip time history and distribution reconstruction allowing tosolve the problem in its general setting. The solution of an auxiliary linearprogramming problem is an essential part of the developed method. Totest the algorithm we use a synthetic displacement function for the faultmodel and perform the inversion.

The slip determination problem can be formalized in the frame ofmathematical programming in the following way

〈c, x〉 → min,A(λ)x = b,x ≥ 0.

(1)

Here x is the unknown vector of amplitudes and residuals (see [2]) andthe vector λ represents the unknown rakes. Note that the displacementfield models can be different but the mathematical formalization is alwaysthe same. If we fix the rake vector λ, problem (1) becomes a linearprogramming problem. This observation is the key to an effective solutionof problem (1). It turns out that the gradient of the minimized functional〈c, x〉 with respect to λ can be calculated in terms of the solution to thelinear programming problem dual to (1).

The following algorithm describes the process.

Algorithm:

Given λ0, ∆ > 0, and ǫ > 0.for k = 0, 1, 2, . . .

Step 1. Solve linear programming problem (1) with λ = λkand obtain xk.

52

Step 2. Obtain search direction λk and a step δk > 0.if δk‖λk‖ < ǫ breakelse

Step 3. Set λk+1 = λk + δkλk.end (for)

The second step of the algorithm is not trivial. The derivative iscalculated using the dual linear programming problem. The latter has avery specific form:

〈c, x〉 → min,Bx ≤ 0,−1 ≤ (x)i ≤ 1, i = 1, n,

(2)

where B is an (m× n)-matrix with m < n. This special structure of thedual problem allows one to effectively find an admissible vertex.

This research is supported by the Portuguese Foundation for Science and

Technologies (FCT), through the project PTDC/CTE-GIN/82704/2006.

References

1. K. Aki, P. Richards. Quantitative Seismology, 2nd edition, University ScienceBooks, Sausalito, (2002).

2. S. Das, B.V. Kostrov. “Inversion for slip rate history and distribution onfault with stabilizing constraints: Application to the 1986 Andreanof Islandsearthquake,” J. Geophys. Res., 95, 6899-6913, (1990).

3. S. Das, P. Suhadolc, B. V. Kostrov. “Realistic inversions to obtain grossproperties of the earthquake faulting process,” Tectonophysics, 261, 165-177,(1996).

4. S.C. Larsen, C.A. Schultz. “ELAS3D, 2D/3D Elastic Finite-Difference WavePropagation Code,” Lawrence Livermore National Laboratory, UCRLMA-121792, (1995).

5. C. Lawson, R. Hanson. Solving Least Squares Problems, Prentice-Hall (seriesin automatic computation), New Jersey, (1974).

6. B.V. Kostrov, S. Das. Principles of Earthquake Source Mechanics, Cam-bridge University Press, Cambridge, (1988).

7. W. Suzuki, S. Aoi, H. Sekiguchi. “Rupture process of the 2008 NorthernIwate Intraslab earthquake derived from strong-motion records,” Bull. Seism.Soc. of America, 99, 2825-2835, (2009).

8. A.N. Tikhonov, V. Ya. Arsenin. Solutions of Ill-Posed Problems, Winston& Sons, New York, (1977).

53

New Integer Programming formulations for theKidney Exchange Problem

Miguel Constantino1, Xenia Klimentova2, Ana Viana3

1 Faculdade de Ciencias da Universidade de Lisboa, Lisboa, Portugal;

[email protected],3 Instituto de Engenharia de Sistemas e Computadores, Porto, Portugal;

[email protected], [email protected]

Problem statement

Kidneys are the most demanded organs for transplantation, but find-ing a suitable kidney can be difficult because of their scarcity and of bloodand/or tissue incompatibility between donor and patient. In recent yearsKidney Exchange Programs [1,2,4,5,6,7] brought a new opportunity forpatients who have someone willing to donate him/her a kidney but, be-cause they are not physiologically compatible, the transplantation cannotbe performed. The Kidney Exchange Problem (KEP) appears within theframe of these programs.

The problem can be represented as a directed graph G(V,A), wherethe set of vertices V is the set of incompatible donor-patient pairs. Twovertices i and j are connected by arc (i, j) ∈ A if a donor from pair i iscompatible with the patient of pair j. A weight wij can be associated toeach arc; in the simplest case wij equals to 0 or 1 ∀(i, j) ∈ A. A feasible“kidney exchange” in the whole graph is defined by a set of vertex-disjointcycles and the objective associated to KEP is to determine the exchangethat maximizes the overall number of transplants.

If there is no bound on the number of pairs that can be involved in acycle, the problem turns into an assignment problem and can be solved inpolynomial time. However, this problem is not of practical interest. First,because all operations in a cycle have to be performed simultaneously,the number of personnel and facilities needed for simultaneous operationsbring several logistics problems. Second, because last-minute testing ofdonor and patient can elicit new incompatibilities that were not detectedbefore causing the donation and all possible exchanges in that cycle to becanceled, it is preferred that cycles are of limited size. Summarizing theKEP can be formulated as follows:

54

Find a maximum weight packing of vertex-disjoint cycles with lengthat most k.

In case k = 2 the problem is a maximum cardinality matching problemwhich can be solved in polynomial time [2,9], but when k ≥ 3 and boundedthe problem is NP-complete [1,7].

Extended edge formulations – new compact formulations

There are two known Integer Programming (IP) models which havebeen proposed independently in [1] and [10] — the so called edge and cycleformulations. Unfortunately, both of these formulations have exponentialnumber constraints or variables, which can become a bottleneck for largescale problems defined on high density graphs. In [1] Column Generationwith Branch-and-Bound is implemented for the cycle formulation. Theauthors report very good results and claim that they where able to solveproblems with up to 10000 vertices in a graph. They also show that thecycle formulation dominates the edge formulation. This paper presentsa new compact formulations for the problem, i.e. a formulations whereboth the number of variables and constraints grows polynomially with thenumber of pairs |V |.

In the first formulation, we consider edge formulation variables xij foreach arc (i, j) ∈ A in the graph G = (V,A): xij = 1 if donor of pair i givesthe kidney to a patient of pair j. Additional assignment variables are usedto prevent exponentional number of constraints in the edge formulation.Let L be an upper bound on the number of cycles in any solution. Suchan upper bound is e.g. L = |V |, because each vertex can participate in atmost one cycle, but more refined bounds could be considered. The cyclesin the solution can be represented by an index l with 1 ≤ l ≤ L. Considerthe following assignment variables: yli = 1 if and only if node i belongs tocycle l.

To avoid multiplicity of solutions an order is imposed in cycles: acycle l in the solution must have node l, and any other nodes must havean index larger than l. Moreover, variables yli can be eliminated whenl and i cannot be in the same cycle. Thus let dlij denote the shortestpath (in terms of number of arcs) between vertices i ∈ V and j ∈ V ingraph G, where i, j ≥ l. For each vertex l ∈ V , build the set of verticesV l = i ∈ V |i ≥ l and dlli+d

lil ≤ k. Denote by L the set of indices l such

that V l 6= ∅. One can write the following formulation with polynomialnumber of constraints.

55

max∑

(i,j)∈Awijxij (1)

∑

j:(j,i)∈Axji =

∑

j:(i,j)∈Axij ∀i ∈ V (2)

∑

j:(i,j)∈Axij 6 1 ∀i ∈ V (3)

∑

i∈V l

yli 6 k ∀l ∈ L (4)

∑

l∈Lyli =

∑

j:(i,j)∈Axij ∀i ∈ V l (5)

yli + xij 6 1 + ylj ∀(i, j) ∈ A, ∀l ∈ L (6)

yli 6 yll ∀i ∈ V l, l ∈ L (7)

xij , yli ∈ 0, 1 ∀i ∈ V l, ∀l ∈ L, ∀(i, j) ∈ A

The objective function (1) maximizes the weighted sum of the ex-change. Constraints (2) assure that the number of kidneys received by pa-tient i is equal to the number of kidneys given by donor i. Constraints (3)guarantee that a donor can only donate one kidney. Constraints (4) modelcardinality of feasible cycles. Constraints (5) ensure that node i is in acycle (

∑j:(i,j)∈A xij = 1) if and only if there is an assignment of i to some

l (∑

l yli = 1) . Constraints (6) state that if node i is in cycle l (yli = 1)

and donor i gives a kidney to recipient j (xij = 1) then node j must alsobe in cycle l (ylj = 1). Constraints (7) guarantee vertex l to appear incycle l in the solution.

In the second formulation we consider the same idea of representingcycles by extra indices l but they are introduced in the decision variablesxij . The problem variables will be xlij if arc (i, j) is selected to be in a

cycle l. In addition to the elimination procedures for variables yli whichwere implemented above, one can also eliminate variables xlij . If there isno cycle of size at most k containing node l and an arc (i, j), ∀j, withl < i, j, then variable xlij can be set to zero or eliminated from the model.

Summarizing, the application of the elimination procedures lead to theconstruction of a subgraph Gl = (V l, Al) for each index l ∈ L, where V l

56

was defined above and Al = (i, j) ∈ A | i, j ∈ V l and d(l, i)+1+d(j, l) ≤k. Then the whole reduced graph is G = (V , A), where V =

⋃l∈L

V l,

A =⋃l∈L

Al. With this notation one can write the reduced extended edge

formulation as follows (xlij will be built over A).

max∑

(i,j)∈Awijx

lij , (3)

∑

j:(j,i)∈Al

xlji =∑

j:(i,j)∈Al

xlij ∀i ∈ V l, ∀l ∈ L, (4)

∑

l∈L

∑

i:(i,j)∈Al

xlij 6 1 ∀j ∈ V , (5)

∑

(i,j)∈Al

xlij 6 k ∀l ∈ L, (6)

∑

j:(i,j)∈Al

xlij 6∑

j:(l,j)∈Al

xllj ∀i ∈ V l, ∀l ∈ L, (7)

xlij ∈ 0, 1 ∀(i, j) ∈ A, ∀l ∈ LThe objective (3) is to maximize the total weight of arcs in the set of

all subgraphs of the graph. Constraints (4) guarantee that in each cycle lthe number of kidneys received by patient i is equal to number of kidneysgiven by donor i. Constraints (5) make sure a donor donates only once.Constraints (6) state that in each cycle l a maximum number of k edgesis allowed. Constraints (7) ensure that whenever an arc (i, j) is in cyclel, there is a cycle containing a node with index l.

Conclusions

Computational experiments where carried out to compare the pro-posed and know formulations in terms of time needed to find an optimalsolution and of the integrality gaps for upper bounds of linear relaxationsof the models. CPU times and bounds were obtained with CPLEX 12.1.A wide variety of test instances was considered, with different graph densi-ties. According to the results we can conclude that the non-compact cycleformulation can be very effective for low density graphs with small values

57

of k. However the importance of considering and using compact formula-tions becomes obvious for larger values of k, especially for problems withdenser graphs.

This work is financed by the ERDF European Regional Development Fund

through the COMPETE Programme (operational programme for competitive-

ness) and by National Funds through the FCT Fundacao para a Ciencia e a

Tecnologia (Portuguese Foundation for Science and Technology) within project

”KEP - New models for enhancing the kidney transplantation process /FCT

ref: PTDC/EGE-GES/110940/2009.“

References

1. A. Blum D. J. Abraham and T. Sandholm. “Clearing algorithms for barterexchange markets: Enabling nationwide kidney exchanges,” Proceedings ofthe 8th ACM conference on Electronic commerce, June 13-16, 2007, pp. 295–304.

2. L. Dorry, S.E. Gentry, D.S. Warren et al. “Kidney paired donation andoptimizing the use of live donor organs,” JAMA, 293, No. 15, 1883–1890(2005).

3. J. Edmonds. “Paths, trees, and flowers,” Canadian Journal of Mathematics,17, 449–467 (1965).

4. G. Thiel, P. Vogelbach, L. Gurke et al. “Crossover renal transplantation:hurdles to be cleared!” Transplant Proc, 33, 811–816 (2001).

5. B.J. Haase-Kromwijk, F.H. Class, W.A. Weimar, M. De Klerk,M.D. Witvliet. “Highly efficient living donor kidney exchange program forboth blood type and crossmatch incompatible donor-recipient combinations,”Transplantation, 82, No. 12, 1616–1620 (2006).

6. K.S. Lee, J.Y. Kwak, O.J. Kwon. “Exchange-donor program in renal trans-plantation: a single-center experience,” Transplant Proc, 31, 344–345 (1999).

7. D.F. Manlove, P. Biro, R. Rizzi. “Maximum weight cycle packing in optimalkidney exchange programs,” Technical Report TR-2009-298. Department ofComputing Science, University of Glasgow, 2009.

8. W.H. Marks et al. “Domino paired kidney donation: a strategy to make bestuse of live non-directed donation,” The Lancet, 368, No. 9533, 419–421(2006).

9. G.N. Nemhauser, L.A. Wolsey. Integer and Combinatorial Optimization, AWiley-Interscience Publication, New-York, 1999.

10. T. Sonmez, A.E. Roth, M.U. Unver. “Efficient kidney exchange: Coincidenceof wants in markets with compatibility-based preferences.”The American Eco-nomic Reviw, 97, No. 3, 828–851, (2007).

58

Identification of parameters in model of watertransfer in soil

V.V. Dikusar1, E.S. Zasukhina2

1 Computing Center RAS, Moscow, Russia; [email protected] Computing Center RAS, Moscow, Russia; [email protected]

A one-dimensional model of vertical water transfer in soil is considered.We assume that soil is an isothermal porous homogeneous medium. Inthat case a water transfer can be described by one-dimensional nonlinearparabolic equation.

Consider following initial-boundary value problem:

∂θ

∂t=

∂

∂z

(D(θ)

∂θ

∂z

)− ∂K

∂z, (z, t) ∈ Q,

θ(z, 0) = ϕ(z), z ∈ (0,L),θ(L, t) = ψ(t), t ∈ (0, T ),∂θ

∂t

∣∣∣∣z=0

=

(D(θ)

∂θ

∂z−K

)∣∣∣∣z=0

+R(t)− E(t), t ∈ (0, T ),

θmin ≤ θ(0, t) ≤ θmax, t ∈ (0, T ).

Here, z is the coordinate; t is time; θ(z, t) is humidity at point (z, t); Q =(0, L)× (0, T ); ϕ(z) and ψ(t) are given functions; D(θ) is the coefficient ofdiffusion; K(θ) is the hydraulic conductivity; R(t) is precipitation; E(t) isevaporation; θmin and θmax are minimal and maximal values of humidityrespectively.

According to [1]-[2] D(θ) and K(θ) are given in the form:

K(θ) = K0S0.5[1− (1− S1/m)m]2,

D(θ) = K01−m

αm(θmax − θmin)S0.5−1/m ×

×[(1− S1/m)−m + (1− S1/m)m − 2],

where S =θ − θmin

θmax − θmin, and K0, α, m are some parameters.

We call this problem a direct problem.Sometimes it is difficult and expensive to determine the parameters

α and m experimentally. Our aim is to determine them as a result ofsolution of inverse problem.

59

Suppose that a function θ(z, t) is defined in Q0 ⊂ Q. This function canbe interpreted as the experimental data. Suppose that the set Q0 consistsof finite number of elements q, q = (z, t). Consider following problem:determine α and m so that the solution θ(z, t) of the direct problem is

close to given function θ(z, t) in Q0. More precisely, the optimal controlproblem is to determine uopt = (αopt,mopt) and corresponding optimalsolution θopt(z, t) such that functional

J(u) =1

2

∑

q∈Q0

ξ(q) · [θ(q)− θ(q)]2

is minimized. Here ξ(q), q ∈ Q0, are weight multipliers such that

ξ(q) ≥ 0, q ∈ Q0,∑

q∈Q0

ξ(q) = 1.

In order to solve optimal control problem numerically it is necessary topass to its discrete analogue. Let us divide the time interval (0, T ) and thespace interval (0, L)) into N and I subintervals respectively. The lengthsof these subintervals are τ = T/N and h = L/I respectively. Considerfollowing finite-difference approximation of the direct problem:

θn+1i − θni

τ=

1

h

(Dni+1/2

θn+1i+1 − θn+1

i

h−Kn

i+1/2−

− Dni−1/2

θn+1i − θn+1

i−1

h+Kn

i−1/2

),

1 < i < I, 0 ≤ n < N,

θ0i = ϕi, 0 ≤ i ≤ I, θnI = ψn, 1 ≤ n ≤ N,where θni , D

ni+1/2, K

ni−1/2 are values of the functions θ(z, t), D(θ(z, t)),

K(θ(z, t)) at the points (ih, nτ), ((i+ 1/2)h, nτ), ((i− 1/2)h, nτ) respec-tively. On the left boundary

θn+10 − θn0

τ=

2

h

(Dn

1/2

θn+11 − θn+1

0

h−Kn

1/2 +Rn+1 − En+1

),

0 ≤ n < N.

Here Rn+1, En+1 are values of R(t) and E(t) at t = (n+ 1)τ .

60

Then the discrete analogue of the direct problem has the form:

Φn0 = −(1

τ+

2

hDn−1

1/2

)θn0 +

2

hDn−1

1/2 θn1+

+1

τθn−10 +

2

h

(−Kn−1

1/2 +Rn − En)= 0,

θmin ≤ θn0 ≤ θmax, 1 ≤ n ≤ N,

Φni =1

h2Dn−1i−1/2θ

ni−1 +

1

h2Dn−1i+1/2θ

ni+1−

−1

τ+

1

h2

(Dn−1i +Dn−1

i−1/2

)θni +

+

θn−1i

τ+

1

h

(Kn−1i−1/2 −Kn−1

i

)= 0,

1 ≤ i ≤ I − 1, 1 ≤ n ≤ N,

ΦnI = θnI − ψn = 0, 1 ≤ n ≤ N,

θ0i = ϕi, 0 ≤ i ≤ I.

(1)

Suppose that Q0 is a set Q0 = (z, t) : z = ih, t = klτ, where0 ≤ i < I, 1 ≤ l ≤ M , M = [N/k] (M is floor of N/k), k is some naturalnumber, k ≥ 1. Then the functional has a form:

W (u) =1

2

I−1∑

i=0

M∑

l=1

ξli · (θni − θni )2,

where n = kl. The discrete optimal control problem is to find optimalcontrol uopt = (αopt,mopt) and correspondent optimal solution of directproblem (1) such that the functional W (u) is minimized.

The system of equations (1) is solved from top to bottom, from firsttime layer to N -th one. On the each time layer the solution is determinedby sweep method. After obtaining θni , 0 ≤ i ≤ I, the value of θn0 iscompared with θmin and θmax. If θn0 < θmin or θn0 > θmax then θn0 is setto θmin or to θmax respectively. Then calculations on the n-th layer arerepeated.

To solve optimal control problem we determine gradient of W . In-troduce the following notations: ΦT = [Φ1

0, ...,Φ1I , ...,Φ

N0 , ...,Φ

NI ], θT =

[θ10 , ..., θ1I , ..., θ

N0 , ..., θ

NI ]. According to methodology of fast automatic dif-

ferentiation [3] the relations for determining the gradient of the functional

61

W are of the form:dW/du =Wu +Φ⊤

u p, (2)

Wθ +Φ⊤θ p = 0n. (3)

The formula (3) of gradient dW/du contains Lagrange multiplier pdetermined by solving system (2). This system is linear with respect top. The basic matrix Φ⊤

θ of the system (2) can be considered as a N ×Nmatrix of block elements. Each block has a size I × I. This block matrixis a upper bidiagonal matrix, i.e. all its nonzero blocks are concentratedon the main diagonal and on the closest diagonal above the main one. Itis clear that system (2) can be split into N systems which can be solvedsequentially from bottom to top. Each of these systems is solved by sweepmethod.

This approach was applied to numerical solution of parameter identi-fication problem with following values of input parameters:

K0 = 100 cm/d, L = 100 cm, T = 30 d,

ϕ(z) = 0.3, z ∈ (0, L),

ψ(t) = 0.3, E(t) = 0, t ∈ (0, T ),

R(t) =

0.1, t ∈ (0, T/2),0, t ∈ (T/2, T ).

θmin = 0.05, θmax = 0.5,

ξ(q) =1

I ·M , q ∈ Q0.

The initial-boundary value problem was considered in dimensionless form.The calculations were carried out on a grid with I = 10 and N = 30.Firstly the direct problem was solved with parameters α = 0.01 andm = 0.2. Obtained function θ(z, t), (z, t) ∈ Q0, was considered as the

prescribed function θ(z, t). Then solving optimal control problem we de-termine uopt = (αopt,mopt). We started with αinit = 0.03 andminit = 0.3.Numerical optimization was executed by gradient method. The gradientwas determined by the described technique. Numerical experiments wereperformed with k = 1, 2, 3, 4. The results obtained are presented in thefollowing table.

62

k αopt mopt Value offunctional

Number ofiterations

1 1.0000004 · 10−2 2.0000001 · 10−1 1.49 · 10−16 7242 9.9999699 · 10−3 1.9999991 · 10−1 4.77 · 10−15 6123 9.9999843 · 10−3 1.9999995 · 10−1 8.77 · 10−16 12024 9.9999308 · 10−3 1.9999979 · 10−1 1.16 · 10−14 1551

Obviously with increasing k the optimization process becomes moredifficult, and obtained parameters αopt and mopt differ more from truevalues α = 0.01 and m = 0.2,


(project no. 10-08-00624 and project no. 11-07-00201).

References

1. Y. Mualem.. “A new model for predicting the hydraulic conductivity of un-saturated porous media,” Water Resour. Res., No. 12, 513–522 (1976).

2. M.Th. Van Genuchten. “A closed form equation for predicting the hydraulicconductivity of unsaturated soils,” Soil. Sci. Soc. Am. J., 44, 892–898(1980).

3. Yu.G. Evtushenko. “Computation of exact gradients in distributed dynamicsystems,” Optimization methods and software, 9, 45–75 (1998).

Improvement Technology for the Accuracy ofSolution of Unconstrained Argument Problems

Anna Dorjieva1

1 Institute for System Dynamics and Control Theory of SB RAS, Irkutsk,


Argument problems form one of the most complicated classes of ex-tremal problems. As is mentioned by some experts [1], today there are noreliable methods for numerical solution of general argument problems.

In this paper we consider two approaches for improvement of solu-tions of unconstrained argument problems. The first approach is based

63

on a specific transformation of the original cost function to an auxiliaryone with the same points of minimum. The second approach refers tothe so-called “Cauchy method”, which reduces our extremal problem to asearch for a numeric solution of a system of ordinary differential equations(ODE). The mentioned approaches are specified by means of special com-putational techniques. Results of numerical experiments are given. Ourexperimental collection contains a family of Skokov-Nesterov functions,also called “generalized Rosenbrock functions” [1], along with continu-ously differentiable and non-differentiable functions proposed recently byY.E. Nesterov [2]:

f(x) =n−1∑

i=1

(1 − xi)2 + c(xi+1 − x2i )2, c = 100, (1)

f(x) =n−1∑

i=1

|1− xi|+ c|xi+1 − |xi||, c = 100, (2)

f(x) =1

4(1− x1)2 +

n−1∑

i=1

(1 + xi+1 − 2x2i )2, (3)

1

4|1− x1|+

n−1∑

i=1

|1 + xi+1 − 2|xi||. (4)

As is well known, the most of local optimization algorithms do not provethemselves effective in minimization of functions of the form (3)-(4), evenin the case of relatively small dimensions. By a “right solution” we meana vector, whose deviation from the optimal value (1, 1 . . . , 1) is less then1% in all variables. Given (−1, 1,−1, 1, . . . , 1) as an initial estimate, theoptimal value of the cost function is equal to zero. In what concernsoptimization of non-smooth Nesterov functions, we apply the well-knownsmoothing method |y| ≈

√y2 + ε2, and an approximation of the original

problem by a sequence of smooth problems depending on a parameter ε.Consider the following optimization problem:

f(x) −→ min, x ∈ En. (5)

Assume that f(x) is a unimodal function vanishing at points of its exactminimum. It is easy to see that the point of global minimum of the

function f(x) coincides with one of the function g(x) = 1 − 1

1 + f(x).

64

Introduce the function F (x) = f(x)+αg(x), where α is a scalar parameter,and consider optimization problem

F (x) −→ min, x ∈ En. (6)

Clearly, optimal solutions of problems (5) and (6) coincide.To solve problem (6) numerically we apply method MSBH (see, e.g.,

[5]), which is implemented in OPTCON-A solver (see [4]).The results of the numerical experiments are presented in Tables 1–

3. Here, # denotes a number of a test function, N is a dimension,f rec is an obtained estimation to the value of the objective function,∆x = max

i=1,...,N(|xrec − 1|) is the maximal deviation in arguments from

the components of an optimal vector.

Table 2: MSBH, the original problem

# N f rec x1 17 5, 66366E − 14 1E − 022 16 1, 54296E − 15 1E − 023 9 4, 04784E − 16 1E − 024 10 8, 25361E − 18 1E − 03

Table 3: MSBH, the reduced problem

# N f rec x α1 26 1, 08420E − 13 1E − 02 1, 00E + 062 30 7, 36295E − 17 1E − 03 1, 00E + 073 16 4, 74014E − 22 1E − 02 1, 00E + 104 16 2, 30513E − 19 1E − 02 1, 00E + 08

Note that in our numerical experiments the values of the parameter αwere given, though they were particular for different dimensions. In whatconcerns the setup of the parameter ε (in the case of non-differentiable

65

functions), we simply put ε = 1E − 12. By varying α and ε in a more so-phisticated way, we succeeded in obtaining “right solutions” for problemsof relatively high dimensions in a series of experiments.

Another approach for improving of the accuracy of solutions of un-constrained argument problems consists in application of a computationaltechnique with a qualifying local search algorithm. According to this tech-nique, the original problem is reduced to the following Cauchy problemfor the system of ODE:

y = −∇f(y), t ∈ [t0, t1],y0 = y(t0).

(7)

Given yk we set yk+1 = yk + hk · ∇f(yk), where hk defines, as usual,the kth “step” of our numerical method, and ∇f(yk) is calculated bymeans of a finite-difference scheme. The least of the values fk = f(yk),k = 1,K, is said to be numerical solution of the original problem, whilethe corresponding values yk are called the points of minimum of the costfunction.

Table 4: The Cauchy method

N f rec x1 24 2, 00759E − 12 1E − 022 28 6, 36721E − 15 1E − 023 14 4, 58770E − 13 1E − 034 16 5, 53190E − 15 1E − 02

Our numerical experiments show that the proposed technique of theproblem transformation to an “inverted” one allows us to increase con-siderably (plus 6 or more variables) the dimension of “rightly solved”problems. We also analyzed the dependence of the “quality” of the ob-tained solutions on values of the parameter α. It was found out that byapplying the qualifying local search, one can increase the dimension ofrightly solved problems in 4 or more variables, depending on values ofhk. Note that in all the problems we solved numerically, the deviation inarguments of the objective function from their optimal values monotoneincrease as the number of a variable raises. Finally we conclude that the

66

proposed computational technologies allows one to increase appreciablythe accuracy of solutions of some unconstrained argument problems.

The author was partially supported by the Russian Foundation for Basic

Research (project no. 12-01-00193).

References

1. B.T. Polyak. Introduction to Optimization, Optimization Software, New York(1987).

2. Yu.E. Nesterov. Introduction to Convex Optimization, MCCME, Moscow(2010).

3. V.K. Isaev, V.V. Sonin. “A new approach to the problem of approximationand its application to the variational and minimax problems,” Proceedings ofTsAGI, 1, No. 1646, 3–23 (1975).

4. A.Yu. Gornov. The computational technologies for solving optimal controlproblems, Nauka, Novosibirsk (2009).

5. M.A. Posypkin. “Searching for minimum energy molecular clusters: methodsand distributed software infrastructure for numerical solution of the problem,”Vestnik of NNSU, 1, No. 1, 210–219 (2010).

On optimal stabilization with respect to a part ofvariables for multiply connected controlled systems

Olga Druzhinina1, Natalia Petrova2

1 Dorodnicyn Computing Center of RAS, Russia; [email protected] Dorodnicyn Computing Center of RAS, Russia; [email protected]

For nonlinear controlled systems methods of optimal stabilization areconsidered in [1]–[5] and other works. In this work we suggest a methodfor solving the problem of optimal stabilization to respect to a part of vari-ables for multiply connected nonlinear controlled dynamic systems givenin the form of nonlinear systems of the ordinary differential equations.The decision of the specified problem reduces to the solving of a problemof optimal stabilization in the sense of V.V. Rumyantsev [1], [2].

67

V.V. Rumyantsev in [1] proved a theorem about the optimal stabi-lization of stable (asymptotically stable) system of differential equationsof perturbed motion with the additional forces under the condition ofminimization of a functional characterizing the quality of control. Thefunctional is given in the form of a definite integral with the upper infi-nite limit. Integrand function of the functional is defined in the proof ofthe theorem, in this case the known Lyapunov function for the system ofdifferential equations of perturbed motion without the control becomesthe optimal Lyapunov function for the specified system under the actionof additional forces.

We consider multiply connected nonlinear controlled dynamic system

dxsdt

= fs(t, xs, ulocs )+Fs(t, x, u

globs ) ≡ Φs(t, x, u

locs , uglobs ), s = 1, q, (1)

where x =(xT1 , ..., x

Tq

)T, xs ∈ Rns , Rn1 ⊕ ... ⊕ Rnq = Rn, ulocs (t, 0) = 0,

uglobs (t, 0) = 0, Φs(t, 0, 0, 0) ≡ 0.It is accepted that right part of system (1) is defined in domain

Ω1 = t, x, ulocs , uglobs : t ≥ t0 ≥ 0, ||x|| < H,||ulocs || <∞, ||uglobs || <∞, 0 < H = const, s = 1, q, (2)

and conditions of existence and uniqueness of solution are satisfied. Letus assume than system (1) can be represented as

dysdt

= Ys(t, ys, zs, ulocs ) +

q∑

j=1

Y1sj(t, y, z)uglobs ,

dzsdt

= Zs(t, ys, zs, ulocs ) +

q∑

j=1

Z1sj(t, y, z)uglobs ,

(3)

where xs = (yTs , zTs )T , x = (yT , zT )T , where ys ∈ Rks , zs ∈ Rns , ks+ms =

ns, s = 1, q. For system (3) domain (2) takes the form

Ω2 = t, x, ulocs , uglobs : t ≥ t0 ≥ 0, ||ys|| < Hs, ||zs|| ≤ ∞,

||ulocs || <∞, ||uglobs || <∞, 0 < H = const, s = 1, q,and each solution is z-extendible.

68

We consider the subsystems of the form

dysdt

= Ys(t, ys, zs, ulocs )

dzsdt

= Zs(t, ys, zs, ulocs ), s = 1, q.

(4)

Further, we will solve the problem of optimal ys-stabilization of multi-ply connected dynamical systems of the form (3), s = 1, q, y = (yT1 , ..., y

Tq )T ,

using the method of Lyapunov vector-functions. In this case the strat-egy of solving the problem of stabilization is that each subsystem must beys-stabilized with the help of local controls ulocs , s = 1, q, i.e. it must be ys-stabilized on the level of subsystems, and then the asymptotic ys-stabilityof interconnected subsystems must be checked. The general scheme of atwo-level stabilization scheme [6] is that the global control uglobs , s = 1, q,is added to the decentralized control in order to weaken the effect of in-terrelated subsystems. In this work the problem of optimal stabilizationof multiply connected system is sold also using a two-level stabilizationscheme with respect to a part of variables.

We consider the case when right parts of (4) can be written in theform

Ys(t, ys, zs, ulocs ) ≡ Y s(t, ys, zs) + b1s(t, ys, zs)u

loc1s ,

Zs(t, ys, zs, ulocs ) ≡ Zs(t, ys, zs) + b2s(t, ys, zs)u

loc2s , s = 1, q,

(5)

where b1s(t, ys, zs) and b2s(t, ys, zs) are matrixes of appropriate dimen-sions, and controls uloc1s and uloc2s are built considering the choice of Lya-punov vector-functions.

It was shown that equilibrium state of system (4) taking into account(5) is uniformly asymptotic ys-stable. In this case system (3) can berepresented in the form

dysdt

= ϕs(t, ys, xs) + Y1s(t, y, z)uglobs ,

dzsdt

= ψs(t, ys, zs) + Z1s(t, y, z)uglobs ,

(6)

where

ϕs(t, ys, zs) = Y (t, ys, zs) + b1s(t, ys, zs)ulocs (t, ys, zs),

ψs(t, ys, zs) = Zs(t, ys, zs) + b2s(t, ys, zs)ulocs (t, ys, zs).

69

For system (6) we consider the problem of optimal stabilization. Cri-terion of quality control we write in the integral form

J =

∞∫

0

w(t, y[t], z[t], uglobs [t])dt, (7)

in this case in the process of solution we define the function w(t, x, u).As optimal Lyapunov function for system (6) we choose the function

V (t, y, z) =

q∑

s=1

αsVs(t, ys, zs),

where αs are positive real constants, Vs(t, xs) are Lyapunov functionswhich guarantee uniform asymptotic ys-stability of systems (4), s = 1, q.

We introduce Krasovsky–Bellman function B(t, x, u, v, uglob) with spe-cial component Ψ(t, y, z) allowing to consider the function w in integral(7) in the form

w(t, y, z, uglob) = Ψ(t, y, z, uglob) +1

2

q∑

s=1

(uglob)T θsuglobs .

According to Rumyantsev theorem function B(t, x, u, v, uglob) is positive-definite with respect to y, and along optimal control (u0s)

glob we have that

∂B

∂uglobs

∣∣∣(u0

s)glob

= 0, s = 1, q. (8)

In result we obtain positive-definite function with respect to y-componentof phase vector of system (3). In this case we can write the criterion ofquality control in the form

J =∞∫t0

(q∑s=1

αsWs(t, ys, zs) +q∑

s,j=1

θsj(u0s)

glob(u0j)glob+

+q∑

s,j=1

θsjuglobs uglobj )dt.

(9)

Thus if (i) for systems (4) ys-stabilyzing (to uniform asymptotic sta-bility) controls ulocs = ulocs (t, xs) exist, s = 1, q, (ii) function Ψ(t, y, z)

70

is positive-definite with respect to vector y of system (3), then controls(u0s)

glob defined from system (8) are the functions solving the problem ofoptimal y-stabilization of system (3) with respect to functional (9).

By the aid of the presented results with applications of results of works[7]–[11] the algorithms of optimal stabilization of multiply connected non-linear controlled systems are constructed.


(project no. 10-08-00826-a).

References

1. V.V. Rumyantsev. ”On optimal stabilization of controlled systems”, Appl.Math. Mech., 34, No. 3, pp. 440–456 (1970).

2. V.V. Rumyantsev, A.S. Oziraner. Stability and stabilization of motion withrespect to part of variables. Nauka, Moscow (1987).

3. N.N. Krasovsky. ”Stabilization problems of controlled motions”. In book:I.G. Malkin, Stability theory of motion. Nauka, Moscow, 1987, pp. 475–517.

4. P. Boudi, L. Gambarella, O.Tenneriello. ”Partial stability of large scale sys-tems”. IEEE Trans. of Automatic Control, AC-2, No. 1, pp. 94–97 (1979).

5. A.S. Andreev, S.P. Bezglasny. ”On stabilization of controlled systems withguaranteed estimate of control quality”. Appl. Math. Mech., 61, No. 1,pp. 44–51 (1997).

6. D. Shil’yak. Decentralized control by complex systems. Mir, Moscow (1994).

7. O.V. Druzhinina, O.N. Masina, E.V. Shchennikova. ”Optimal stabilizationof programmed motion of manipulation systems”. Dynamics of complex sys-tems, 5, No. 3, pp. 58–64 (2011).

8. O.V. Druzhinina, N.P. Petrova, E.V. Shchennikova. ”Optimal stabilizationof multiply connected controlled systems”. Proc. of II International con-ference ”Optimization and applications” (OPTIMA-2011) held at Petrovac,Montenegro, in September 25 – October 2, 2011. University of Montenegro,Dorodnicyn Computing Center of RAS, Moscow, 2011, pp. 55–57.

9. O.V. Druzhinina, E.V. Shchennikova. ”On optimal stabilization problem ofsystems with gomogenious principal parts”, Trans. of System Analysis Insti-tute of RAS, 49(1), pp. 20–25 (2010).

10. O.V. Druzhinina, A.A. Shestakov. ”Generalized direct Lyapunov method forresearch of stability and attraction in general time systems”, Sbornik Math-ematics, 193, No. 10, pp. 17–48 (2002).

11. A.A. Shestakov, O.V. Druzhinina. ”Lyapunov function method for the analy-sis of dissipative autonomous dynamic processes”, Differential Equations, 45,No. 8, pp. 1108–1115 (2009).

71

Numerical experiments on computer model of acerebellum

W. Dunin-Barkowski1, L. Vyshinskiy2

1 Scientific Research Institute for SystemAnalysis RAS, Moscow, Russia;

[email protected] A.A. Dorodnitsyn Computing Center RAS, Moscow, Russia; [email protected]

Computer modeling of neural networks of big dimension becomes theeffective tool in studying of mechanisms of processing of information in ahuman brain. It is connected, on the one hand, with increase in speed ofmodern computers at the expense of application of parallel calculations,and on the other hand with achievements of neurophysiology which isclose to understanding of many mechanisms of work of a brain. New pos-sibilities allow to investigate various schemes and algorithms of processingof information which ”are invented” by the nature and, maybe, to findof them application for the solution of various applied tasks. It is verytempting ideas as real neural structures of live organisms, and first of alla human brain, successfully solve the most difficult problems of forecast-ing, management and control over various situations. In the report theresearches which are carried out by authors in the field of mathematicalmodeling of neural structures of a cerebellum are presented ([1]).

Cerebellum - one of the most studied independent structures of a brain.The cerebellum contains more than all other structures of a brain of neu-rons. He receives a large volume of information from the outside. About40 million nervous fibers are connected by a cerebellum with a bark ofbig hemispheres and approximately as with peripheral nervous system.At a cerebellum rather homogeneous neural structure which speaks aboutexistence of some general mechanism of calculation of millions the param-eters necessary in the solution of various problems of movements controland not only movement. This mechanism as a first approximation in-cludes some stages: processing of arriving entrance information from abark of big hemispheres and peripheral nervous system, control (training)of neural structures of a cerebellum according to teams arriving from theoutside, calculation of values of the parameters necessary for solution ofproblems of control of an organism and their transfer to a bark of bighemispheres or to peripheral nervous system. Problems of calculation of

72

a large number of multidimensional functions in a cerebellum are solvedby means of homogeneous structure with feedback which includes variouslayers of nervous cells. On the drawing (Fig. 1) the block diagram of acerebellum model which is studied in the real work is shown.

Fig. 1

To each layer of cells presented on the chart there corresponds the setof the equations describing behavior of this layer.

M(t) = K(X, t), (1)

TGdGi/dt = −Gi(t) + ΣjgijM j(t), i ∈ 1, .., ng, j ∈ 1, .., nm, (2)

TPdPi/dt = −Pi(t) + Σjpijwij(t)Gj(t), i ∈ 1, .., np, j ∈ 1, .., ng, (3)

TNdNi/dt = −Ni(t) + ΣjnijPj(t), i ∈ 1, .., nn, j ∈ 1, .., np, (4)

TCFdCi/dt = Ψ(Ci(t),ΣjcijNj(t) + Yi(t)), i ∈ 1, .., nc, j ∈ 1, .., nn,

Ψ(C(t), F (t)) = if(C(t)<Ho)then(F (t)− C(t))else(−Ho ∗ δ(t)), (5)

dwij/dt = εCFΣkekiχ(t− τk(t))Gj(t),i ∈ 1, .., np, j ∈ 1, .., ng, k ∈ 1, .., nc, (6)

Z(t) = H(N(t)). (7)

73

Multidimensional function K(X,t) models transformation of informa-tion coming of external structures in signals on mossy fibers of a cere-bellum. By means of this function it is possible to set various ways ofcoding of entrance information. In the report some options of coding areconsidered and compared.

The group of the equations (2) describes work of a layer of granularcells which perceive and will transform signals from mossy fibers. Thelayer of granular cells is the most numerous in cerebellum structures. Thequantity of cells in this layer can reach hundred thousands. Entrances ofgranular cells are connected to mossy fibers and perceive signals comingfrom the outside through these fibers. At granular cages small numberof entrances (5-10). Connections with mossy fibers (a matrix ||gij||) havecasual character, however various hypotheses here can be used.

The equations (3) are the equations of Purkinye’s cells, the main cal-culator in a cerebellum. Purkinye’s cells are multidimensional arithmeticadders of the weighed signals from granular cells. The quantity of en-trances of Purkinye’s cells can make tens of thousands. Conductivity ofcommunications from granular cages to synapse Purkinye’s cells (synapticweight) cope fiber cells. Synaptic weights are the instrument of control offeedback in a cerebellum.

The layer of small nuclear cells (the equations 4) provides feedback instructure from Purkinye’s cells to climbing fiber cells.

The equations (5) describe behavior of climbing fiber cells. Signalscome to climbing fiber cells from Purkinye’s cells through small nuclearcells and external signals which play a role of operating (training) sig-nals models. The climbing fiber cell can be described as pulse systemwhich periodically gives out the impulses correcting synaptic weights onentrances of cells of Purkinye. This process closes feedback in a cerebellumand allows regulating and adjusting its work on performance of variousfunctions.

Processes of change of values of synaptic weights are described by theequations (6). The fiber cell submits an impulse on synoptic knots to themoments of a relaxation of function Ψ. It changes weight for all activecommunications. In the equation (6) τ – the last moment of a relaxationa fiber cell of the connected with concrete synaptic knot. The functionχ(t) sets an impulse form.

74

Multidimensional functionH(N) (7) forms the target information whichis transferring from a cerebellum in a bark of big hemispheres and in pe-ripheral nervous system.

For studying of this model the special research stand was developed.This stand is interactive program for modeling the equations (1) - (7)for various ways of formation of matrixes of communication ||gij||, ||pij||,||nij||, ||cij||, and various values of parameters of the equations.

In the report results of the computing experiments which have beencarried out at this stand will be presented. Entrance variables which handover external information in cerebellum model, are divided into two canals– the X-th and Y. On the channel X-th information gets through mossyfibers and granular cells on Purkinye’s cells. On the channel Y of infor-mation arrives on climbing fiber cells and influences control of synapticweights. Therefore the channel Y is considered as suited ”management”or ”training”. In examples of figure 2 some tasks which decided in exper-iments with cerebellum model are shown.

Fig. 2

75

a) approximation of functions; b) forecasting of numerical sequences; c)modeling of dynamic systems and training of control of dynamic systems.(variable “u” on schemes is the switch of model’s operating modes – train-ing or check)

Program realization of the interactive stand is executed with the helpof “The generator of projects” ([2]).


(project no. 10-07-00206).

References

1. W.L. Dunin-Barkowski, Yu.A. Flerov, L.L. Vyshinsky. “Prognosis of dynam-ical systems behavior based on cerebellar-type neural technologies”, OpticalMemory and Neural Networks, 20, No. 1, 43–58 (2011).

2. L.L. Vyshinsky, I.L. Grinev, Yu.A. Flerov, A.N. Shirokov, N.I. Shi-rokov. “The generator of projects a tool complex for development client-server systems”, Information technologies and computing systems, No 1-2,6–25 (2003).

P -th order methods for solving nonlinear system

Yu.G. Evtushenko1, A.A. Tretyakov1,2,3

1 Dorodnicyn Computing Center RAS, Moscow, Russia;2 System Res. Inst., Polish Acad. Sie, Warsaw, Poland;

3 University of Podlasie, Siedlce, Poland;


For a system of nonlinear equations, we derive a formula for a familyof its approximate solutions, which provides an elementary proof of thetangent direction theorem and the implicit function theorem. Iterativemethods with a local p-order convergence rate are constructed for solvingsystems of equations. An elementary proof of the Lagrange theorem onnecessary extremum conditions in an equality constraint nonlinear pro-gramming problem is given as an application.

76

Given a vector function F (x):

F (x) : Rn → Rm, FT(x) = [f1(x), f2(x), . . . , fm(x)],

where f i(x) : Rn → R1, i = 1, 2, . . . ,m, m ≤ n, F (x) ∈ C3(Rn), we define

the feasible setX = x ∈ R

n | F (x) = 0m. (1)

The rectangular m × n Jacobi matrix Fx(x) and its kernel and image ata point x ∈ R

n are defined as

KerFx(x) = h ∈ Rn | Fx(x)h = 0m,

ImFx(x) = η ∈ Rm | η = Fx(x)y, y ∈ R

n = (KerTFx(x))⊥.

(2)

Let the rank of Fx(x) be m. Then there exists a nonsingular m ×mGram matrix G(x) = Fx(x)F

Tx (x) and a pseudoinverse (right inverse)

F+x (x) = FT

x (x)G−1(x) of Fx(x); i.e., Fx(x)F

+x (x) = Im. Define two

n-dimensional vectors

N(x) = F+x (x)F (x), p(x) = −1

2F+x (x)hFxx(x∗)h

T. (3)

Here, h ∈ Rn and hTFxx(x∗)h is an m-dimensional vector whose ith com-

ponent is hTf ixx(x∗)h.Theorem 1. Let a point x∗ ∈ X be such that the columns of the

matrix FTx (x∗) are linearly independent and h ∈ KerFx(x∗).

Then in Rn there is a family of arcs

x(α) = x∗ + αh+ α2p(x∗), (4)

that issue from x∗ and belong to the set X up to second-order quantities.

Based on formulas (3) and (4), we can construct an iterative methodfor solving the system F (x) = 0m with m ≤ n. For the mapping F (x) tobe approximated up to an O(‖x− x∗‖3) error in (4), we use, as x(α), x∗,αh, and α2p(x∗), the vectors

xi+1, xi, −N(xi), −(1/2)F+x (xi)N

T(xi)Fxx(xi)N(xi),

respectively. As a result, we obtain the iterative method

xi+1 = xi −N(xi)−1

2F+x (xi)N

T(xi)Fxx(xi)N(xi). (5)

77

Theorem 2. Suppose that F ∈ C2(Rn) and there exists a point x∗ ∈X at which the rank of the matrix Fx(x∗) is m.

Then there exists a neighborhood U(x∗) of x∗ such iterative process (5)converges at a cubic rate for any point xi ∈ U(x∗); moreover,

dist(xi+1, X) ≤ c dist(xi, X)3,

where c > 0 is an independent constant.

Note that xk converges to a point x ∈ X so as to satisfy the conver-gence rate estimate

‖xi+1 − x‖ ≤ c ‖xi − x‖3.

Consider the special case of system (1) where F : Rn → Rn. Assume

that the square matrix Fx(x) is nonsingular. Then the computationalformulas simplify, specifically, F+

x (x) = F−1x (x), N(x) = F−1

x (x)F (x),and iterative method (5) becomes

xi+1 = xi − F−1x (xi)

(F (xi) +

1

2NT(xi)Fxx(xi)N(xi)

). (6)

Theorem 3. Suppose that F ∈ C2(Rn) and there exists a point x∗ ∈X, where the matrix Fx(x∗) is nonsingular.

Then iterative method (6) locally converges to x∗ at a cubic rate:

‖xi+1 − x∗‖ ≤ c ‖xi − x∗‖3, (7)

where is a constant.

In the special case of n = m = 1, it follows from (6) that

xi+1 = xi −f(xi)

fx(xi)− f2(xi)fxx(xi)

2f3x(xi)

.

This method for finding the roots of the equation f(x) = 0 was pro-posed in 1838 by Chebyshev [1]. Iterative methods converging faster thanquadratically were studied in [2, 3].

Consider the case p ≥ 3 for solving following system:

F (x) = 0n, x ∈ Rn. (10)

78

Assume that the point x belongs to a sufficiently small neighborhood ofx∗. Then use the Teylor formula we obtain

F (x∗) = 0n = F (x) − F ′(x)h+1

2F (2)(x)[h]2 + · · ·+

+(−1)pp!

F (p)(x)[h]p +On(‖h‖p+1

).

(11)

Here h(x) = x−x∗, [h(x)]p — multivector, On(‖ · ‖p+1) — n-dimensionalvector with (p+1)-order approximation by h of each component, F ′(x) =Fx(x). Suppose that the matrix F ′(x∗) nondegenerate. Then these existsneighborhood Uε of the point x∗ such that F ′(x) is nondegenerate and wedefine vector N(x) = [F ′(x)]−1F (x).

Let us multiple (11) by [F ′(x)]−1 we obtain

x− x∗ = h1(x) + [F ′(x)]−1

[1

2F (2)(x)[h]2 + · · ·+

+(−1)pp!

F (p)(x)[h]p]+On

(‖h‖p+1

),

(12)

where h1(x) = N(x). Denote

x1(x) = x− h1(x).

Omitting the second order terms we obtain from (12)

‖x1(x)− x∗‖ = On(‖x− x∗‖2).

If we take x1(x) as approximate solution to the system (10), then weobtain

F (x1(x)) = f(x)− F ′(x)h1(x) +On(‖x− x∗‖2) = On(‖x− x∗‖2). (13)

Thus vector x1(x) define approximation solution of system (10) with sec-ond order explicitly of ‖x− x∗‖.

In the right side of (12) omit the terms of third order we obtain

x− x∗ − h2(x) +On(‖x− x∗‖3), x2 = x− h2(x), (14)

where

h2(x) = h1(x) +1

2[F ′(x)]

−1F (2)(x)[h1(x)]

2. (15)

79

Denote by x2(x) = x − h2(x). Take place the estimations analogousto (14), (15)

‖F (x2)‖ ≈ On(‖x− x∗‖3), ‖x2(x) − x∗‖ ≈ On(‖x− x∗‖3).

Analogously, omit in (3) the terms of fourth order we find

x− x∗ = h3(x)−On(‖x− x∗‖4), x3(x) = x− h3(x), (16)

where

h3= h1 + [f ′(x)]−1

1

2F (2)(x)[h2]

2 − 1

3!F (3)(x)[h2]

3

and

‖F (x3)‖ ≈ O(‖x− x∗‖4), ‖x3(x) − x∗‖ ≈ O(‖x− x∗‖4).

Continue this process we are coming to following formulas:

x− x∗ = hp(x)−On(‖x− x∗‖p+1), xp(x) = x− hp(x),

where

hp(x) = h1(x) + [F ′(x)]−1

1

2F (2)(x)[hp−1]

2 + · · ·+

+(−1)pp!

F (p)(x)[hp−1]p

.

(17)

Generalize all what has been said above we can assert that will be truefollowing result.

Theorem 4. Suppose that F ∈ Cp+1(Rn,Rn), there exists a pointx∗ ∈ X where the matrix F ′(x∗) is nonsingular, xi ∈ Uε(x∗) and ε > 0sufficiently small.

Then for the point xi+1

xi+1 = xi − [F ′(xi)]−1

F (xi) +

1

2F ′′(xi)[hp−1]

2 + · · ·+

+(−1)pp!

F (p)(xi)[hp−1]p

(18)

80

will be hold the following estimations:

‖F (xi+1)‖ ≤ c ‖xi − x∗‖p+1, ‖xi+1 − x∗‖ ≤ c ‖xi − x∗‖p+1, (19)

where c > 0 independent constant.


(grant No 11-01-00786a) and the Council for the State Support of Leading

Scientific Schools (grant 5264.2012.1).

References

1. P.L. Chebyshev. Collected Works (Akad. Nauk SSSR, Moscow, 1951). Vol. 5,7–25 (in Russian).

2. A.A. Tretyakov A.A. “Necessary and sufficient conditions for optimality ofp-th order”, Zh. Vychisl. Mat. Mat. Fiz., 24, 986–992 (1984).

3. A.A. Denisov, V.G. Karmanov, and A.A. Tretyakov. “Accelerated Newtonmethod for solving functional equations”, Dokl. Akad. Nauk SSSR, 281,1293–1297 (1985).

Some Complexity Results for the Simple AssemblyLine Balancing Problem

Evgeny R. Gafarov1, Alexandre Dolgui2, Alexander Lazarev3

1 Ecole Nationale Superieure des Mines, FAYOL-EMSE, CNRS:UMR6158,

LIMOS, F-42023 Saint-Etienne, France; [email protected] Ecole Nationale Superieure des Mines, FAYOL-EMSE, CNRS:UMR6158,

LIMOS, F-42023 Saint-Etienne, France; [email protected] Institute of Control Sciences of the Russian Academy of Sciences,

Profsoyuznaya st. 65, 117997 Moscow, Russia; [email protected]

Introduction

We consider the simple assembly line balancing problem (SALBP-1) whichis formulated as follows.

81

Given a set N = 1, 2, . . . , n of operations and K stations (machines)1, 2, . . . ,K. For each operation j ∈ N a processing time tj ≥ 0 is defined.The cycle time c ≥ maxtj, j ∈ N is given. Furthermore, finish-startprecedence relations i → j are defined between the operations accordingto an acyclic directed graph G. The objective is to assign each operationj, j = 1, 2, . . . , n, to a station in such a way that:- number m ≤M of stations used is minimized;- for each station k = 1, 2, . . . ,m a total load time

∑j∈Nk

tj does notexceed c, where Nk – a set of operations assigned to a station k;- given precedence relations are fulfilled, i.e. if i→ j, i ∈ Nk1 and j ∈ Nk2then k1 ≤ k2.

A survey on results for NP-hard in the strong sense SALBP-1 is pre-sented, e.g., in [1,2].Partition problem. Given is a set N = b1, b2, . . . , bn of numbers b1 ≥b2 ≥ · · · ≥ bn > 0 with bi ∈ Z+, i = 1, 2, . . . , n, and a number A ∈ Z+

with A <∑

j∈N bj . Is there a subset N ′ ⊂ N such that∑j∈N ′ bj = A?

The worst case running time of B&B algorithms for the well-knownKnapsack Problem is analyzed, e.g., in [4]. In these papers authors chooseto use only special cases, for which it is fairly easy to find an optimal so-lution with a B&B algorithm. However to prove its optimality, almostall feasible solutions should be considered. We use a similar idea to con-struct a special case of SALBP-1 for which each B&B algorithm with nomatter what polynomial time computed Lower Bound has an exponentialrun time. This makes algorithms ineffective for instances with n ≥ 60operations.Modified instance of the Partition problem. Given is a set N =b1, b2, . . . , b2n of numbers b1 ≥ b2 ≥ · · · ≥ b2n > 0 with bi ∈ Z+, i =1, 2, . . . , 2n, and a number A ∈ Z+ with A <

∑j∈N bj . The numbers

bi, i = 1, 2, . . . , 2n are denoted as follows:

b2n = 1, b2i = 2 ·n∑

j=i+1

b2j−1, i = n−1 . . . , 1, b2i−1 = b2i+bi, i = n . . . , 1,

where b1, b2, . . . , bn – numbers from the initial instance. Let A =∑ni=1 b2i+

A. Without lost of generality let us assume A = 12

∑ni=1 bi and as con-

sequence A = 12

∑2ni=1 bi. The question is: ”Is there a subset N

′ ⊂ N

such that∑

j∈N ′ bj = A”? If for the initial instance of the Partition

82

Problem the answer is ”YES” (and the same answer has the modified

instance) then N ′ contains one and only one number bifrom each pair

b2i−1, b2i, i = 1, 2, . . . , n. If the number bi is included in the set N ′

then b2i−1 is included in N ′, otherwise the number b2i ∈ N ′.In the special case of SALBP-1 there are 2n operations. Let w′ =

minw|10w ≥ 2A. Let us

ti = 10w′

+ bi, i = 1, 2, . . . , 2n

and c = 12

∑2ni=1 ti. There are no precedence relations between operations.

It is obvious that if and only if for the modified instance of the Par-tition Problem the answer is ”YES” then the minimal umber of stationsm∗ = 2, otherwise m∗ = 3. As a consequence, if NP 6= P , there is nopolynomial time computed Lower Bound with a relative error equal orless than 3

2 . That means, for any set of polynomial time computed LowerBounds LB1, LB2, . . . , LBX, there is a modified instance of the Parti-tion Problem with an answer ”NO”, for which LBx = 2, i = 1, 2, . . . , X ,although m∗ = 3. For the special case of SALBP-1, any feasible solutionis optimal. However, to prove its optimality almost all feasible solutionsmust be considered.

Let us estimate the possible number of feasible solutions. On the firststation there could be processed at least n−1 operations. Thus, there areat least

(2nn−1

)possible loads of the first station, i.e. the number of feasible

solutions which have to be considered is greater than(

2nn−1

)= n+1

n

(2nn

)≈

n+1n · 2

2n√nπ. To solve such the instance of SALBP-1 with 2n = 60 a computer

must perform more than 260

10 operations. Let us assume that the fastestknown computer performs 230 operations per second, or less than 247

operations per day. Then a run time of an algorithm will be more than213

10 > 800 days! That means there are instances of SALBP-1 for whichany B&B algorithm with polynomial time computed Lower Bounds hasan unappropriate running time.

We can conclude the following. Despite the best known algorithmB&B [3] solves all benchmark instances in less than 1 second per in-stance, known B&B algorithms for SALBP-1 remain exponential and cannot solve some instances with the size n > 60 in an appropriate time.That is why we consider exact algorithms for the general case of theproblem unpromising. Researchers can concentrate on special cases or on

83

essentially new solution schemes.

Maximization of Number of StationsTo propose an essentially new solution scheme for SALBP-1, it is nec-

essary to investigate properties of optimal solutions. We can investigatenot only properties of good solutions to try imitate their character butproperties of poor solutions as well to avoid solutions with their aspects.Here, in contrast to standart SALBP-1, where the number of stations usedshould be minimized we consider an optimization problem with the op-posite objective criteria, in other words the maximization of the numberof stations. The investigation of a particular problem with the maximumcriterion is an important theoretical task [5]. To make the maximiza-tion problem not trivial we assume that all stations (instead the last one)should be maximal loaded, i.e. for two stations m1,m2, m1 < m2 thereis no operation j assigned on the station m2 which can be assigned onstation m1 without violation of precedence constraints or the feasibility’scondition ”total load time of the station does not exceed the cycle time”.Denote the maximization problem by max− SALBP − 1.

Theorem 1. max-SALBP-1 is NP-hard in the strong sense (by re-duction from the 3-Partition Problem).

Theorem 2. max-SALBP-1 is not approximated with an approxima-tion ratio ≤ 3

2 unlike P = NP .An experimental study of maximal number of stations for benchmark

instances published on http://www.assembly-line-balancing.de was done.The results show that the maximal founded deviation mmax−mmin doesnot exceed 20%.

Flat Graph of Precedence RelationsIn [6] authors propose a transformation of graph G of precedence re-

lations to planar one for the well-known Resource-Constrained ProjectScheduling Problem. The same idea can be used for SALBP-1.

Theorem 3. For any instance of SALBP-1 with n operations and vprecedence relations, there exists an analogous instance with a flat graphG′ with n′ operations and v′ relations, where n+ v ≥ n′ + v′.

We obtain an analogous instance from the original one by adding”dummy” operations (with tj = 0) and deleting all the unnecessary rela-tions. According to the well-known Euler’s Theorem, v′ ≤ 3n′− 6 in suchthe planar graph.

84

The number of precedence relations influences running time and thetheoretical complexity of solution algorithms. The number of precedencerelations is estimated by different authors asO(n2) (i.e, the ”Order strength”on http://www.assembly-line-balancing.de is estimated according to thenumber n · (n− 1) of precedence relations). If we consider only instanceswith planar graphs then the number of relations is ≤ 3n − 6, i.e. O(n).So, the fact mentioned in Theorem 3 allows us to reduce the run timeof algorithms (by reduction of unnecessary relations) and estimate thecomplexity exacter.


(projects no. 11-08-01321, 11-08-13121).

References

1. E. Erel, S.C. Sarin, A survey of the assembly line balancing procedures,Production Planning and Control, 9, 414434 (1998).

2. A. Scholl, Balancing and Sequencing of Assembly Lines, Physica Verlag, ASpringer-Verlag Company, 1999.

3. E.C. Sewell, S.H. Jacobson, A Branch, Bound and Remember Algorithmfor the Simple Assembly Line Balancing Problem, INFORMS Journal onComputing, doi 10.1287/ijoc.1110.0462, 1–10 (2011)

4. M.A. Posypkin , I. Kh. Sigal , Speedup estimates for some variants of theparallel implementations of the branch-and-bound method, ComputationalMathematics and Mathematical Physics, 46, N 12, 2189 –2 202 (2006).

5. M.A. Aloulou , M.Y. Kovalyov, M.-C. Portmann, Maximization Problemsin Single Machine Scheduling, Annals of Operations Research, 129, 21 – 32(2004).

6. A.A. Lazarev, E.R. Gafarov, Transformation of the Network Graph ofScheduling Problems with Precedence Constraints to a Planar Graph, Dok-lady Mathematics, 79 N 1, 1-3 (2009).

85

Optimization of a multiple covering of a surfacetaking into account its relief

Shamil Galiev1, Maria Lisafina2, Vitalii Yudin3

1 National Research Technical University; Kazan, Russia; [email protected] National Research Technical University; Kazan, Russia; e [email protected]

3 National Research Technical University; Kazan, Russia;

[email protected]

We propose a numerical technique for optimization of a k-fold (k ≥ 1)covering of a bounded part of a given surface G by some figures taking intoaccount the relief of the surface G. The k-fold covering implies that eachpoint of the surface G is observable from at least k centres of coveringfigures, while the segment connecting the observer with the observableobject contains no points that lie below the surface G. For k > 1 weadditionally require that the distance between centres of figures is notless than some given value. We construct models and propose algorithmsfor determining the minimal possible number of figures and the locationof centres under the indicated conditions and the additional requirement.The obtained numerical results demonstrate the efficiency of the proposedtechnique.

The problem of covering a bounded plane parts by circles of minimalradii or that of covering a bounded plane parts by the least number ofcircles of a given radius widely studied in the literature. This is connectedboth with the mathematical interest to the mentioned combinatorial opti-mization problem and with various applications such as the problem of thechoice of the quantity and the location of various stations (for example,those of cellular communication, medical ambulance, technical services,etc). One can find possible applications of covering problems and theirsolution methods, for example, in [1].

In problems of determining the quantity and location of componentsof forest fire control systems such as Fire Watch [5], Lesnoy Dozor (ForestPatrol) [6], or elements of lidar (LIght Detection and Ranging) systems[2] it is necessary to take into account the lay of land. The problem ofdetermining the quantity and location of video cameras in security systemsis also close (in a sense) to the mentioned problem. Thus, in many casesone encounters covering problems that take into account the observability

86

of the controllable object. One can treat such problems as problems ofcovering the surface taking into account its relief.

In navigation systems or image recognition ones it is necessary to pro-vide a multiple covering by observation zones. Moreover, there is no needin placing more than one station or camera at one and the same point,because this does not help to solve the stated problems. Therefore, onehas to impose an additional constraint on the minimal distance betweencentres of figures.

Let E3 be a three-dimensional Euclidean space with a Cartesian co-ordinate system with axes 0x, 0y, and 0z; let P1 be a bounded domain inthe plane x0y. Assume that a function g(x, y), (x, y) ∈ P1, is Lipschitzon P1 and its Lipschitz constant is known. We denote the surface definedby the function g(x, y) by G. We treat a point p ∈ G as observable froma point (a centre) s ∈ G, if the segment [p, s] contains no points that liebelow the surface G and the length of the segment does not exceed a givenvalue r1. Let us state the following problem.

P r o b l e m Z. Determine the least number of centres sq1, sq2, . . . , sqm,m ≥ 1, and their location on G so as to make any point p ∈ G observablefrom at least k (1 ≤ k ≤ m) centres. Moreover, the distance betweenthe observable point p and the observing centre should not exceed r1, andwith k > 1 the minimal distances between these centres should be not lessthan the given value r2.

The set of points p ∈ G observable from sj form some figure Kj in G;we treat sj as the centre of Kj , 1 ≤ j ≤ m. If G is a plane, this figurerepresents a circle of radius r1.

We introduce a discrete analogue of the covering problem for G andtreat its solution as an approximate solution of the initial problem. OnP1 we construct a rectangular grid with some step ∆x along axes 0x and0y. Let pp1, . . . , ppn be the set of nodes of the constructed grid on P1

(ppj ∈ P1, 1 ≤ j ≤ n) and let points ppj have coordinates xj , yj , zj(zj = 0), 1 ≤ j ≤ n. Let points pj have the same coordinates xj , yj aspoints ppj , and let the coordinate zj equal zj = g(xj , yj)+αj, αj ≥ 0, 1 ≤j ≤ n. Now instead of the covering of the whole surface G we consider thecovering of the set of pointsGn = p1, p2, . . . , pn. We assume that centresare located at some points of the set S = s1, s2, . . . , sn, where pointssj have the same coordinates xj , yj as points ppj , and the coordinate zequals zsj = g(xj , yj) + βj , βj ≥ 0, 1 ≤ j ≤ n. We introduce values αj

87

and βj , 1 ≤ j ≤ n, in order to take into account the fact that an observer(for example, an antenna or a video camera) can be located above thesurface G, as well as the fact that the observable object can be locatedabove G.

Let d(s, t) be the Euclidean distance between points s, t ∈ E3, and letr1 and r2 be some given positive values. For the set of points pi and sjwe calculate distances dij = d(pi, sj), 1 ≤ i, j ≤ n. We find values aij ,1 ≤ i, j ≤ n, by the following procedure.

P r o c e d u r e W1:1. Choose points pi and sj in sets Gn and S, respectively, 1 ≤ i, j ≤ n.2. If d(pi, sj) > r1, then we set aij = 0 and choose the next pair of

points pi and sj in sets Gn and S, respectively, 1 ≤ i, j ≤ n.3. If d(pi, sj) ≤ r1, then we find out whether at least one point of the

segment [pi, sj ] is located below the surface G. If there are no such points,then we set aij = 1, otherwise we do aij = 0 and choose the next pair ofpoints pi and sj in sets Gn and S, respectively, 1 ≤ i, j ≤ n.

4. If all possible points pi and sj of sets Gn and S, respectively,1≤ i, j ≤ n, are considered already, then stop.

As a result of the implementation of Procedure W1, we find all valuesof aij , 1≤ i, j ≤ n,which form an n × n matrix A = (aij) consisting ofzeros and units.

P r o c e d u r e W2.1. Choose points si and sj in the set S, 1 ≤ i, j ≤ n.2. If d(si, sj) ≤ r2, then we set: a) bii = mi+1, where mi is a number

of point sj , 1 ≤ j ≤ n, for which d(si, sj) ≤ r2; b) bij = 1 if i 6= j;otherwise (for d(si, sj) > r2) we do bij = 0 choose the next pair of pointssi and sj in the set S, 1 ≤ i, j ≤ n.

3. If all possible points si and sj of the set S, 1 ≤ i, j ≤ n, areconsidered already, then stop.

As a result of the implementation of Procedure W2, we find all valuesof bij , 1 ≤ i, j ≤ n, which form an n × n matrix B = (bij) consisting ofzeros, units and mi + 1, 1 ≤ i ≤ n.

Introduce the following denotations:

tj =

1, if the center of the jth figure is located at the point sj ,0 otherwise;

88

t = (t1, t2, . . . , tn)T , while I and V are n-dimensional vectors, namely,

I = (1, 1, . . . , 1)T and V = (k, k, . . . , k)T .Let us state the problem

min

n∑

i=1

ti : At ≥ V, t ∈ 0, 1n. (1)

Problem (1) is the well-known covering problem, and in this case itmeans the determination of the least quantity of figures that form a cov-ering of the set Gn, taking into account the relief of the surface G, andthe determination of centres of covering figures on the set S. In our talkwe describe the solvability conditions for problem (1), using the resultsobtained in [4] and [3]. We also consider other variants of the problem,including the weighted one.

The problem that takes into account the additional requirement

minimal distances between centres is not less than r2 (2)

is stated analogously. In order to take into account the conditions (2),one has to impose the constraint Bt ≤ M , where M = (m1 + 1,m2 +1, . . . ,mn+1)T . Note that we have not succeeded in finding such a con-dition for the covering problem in the related literature. We introducethis constraint for a k-fold (k > 1) covering, but, if necessary, one canintroduce it even for k = 1.

For the realization of the step 3 of Procedure W1 we used the Lipschitzproperty of the function g(x, y). Then for solving problem (1) we applieda well-known method based on the relaxation, the construction of the coreproblem, and its solution by one of known methods.

For solving Problem Z we developed a computer program (using CPLEX-11.2) and obtained numerical results. Consider the case when the domainP1 is the unit square S. Assume that the function g(x, y) = 0, i.e., thesurface G coincides with S. Note that in the k-fold covering problem,condition (2) can be fulfilled with certain values of r2. For example, inthe 2-fold covering of S by five circles, it is possible that the minimal ra-dius of circles r1 ≈ 0.527, while condition (2) is fulfilled with r2 ≤ 0.336;the 3-fold covering of S by eight circles takes place with r1 ≈ 0.518, whilecondition (2) is fulfilled with r2 ≤ 0.264. But in many cases one shouldintroduce condition (2) by using the matrix B. The location of four circles

89

of the radius√2/4 that form an 1-fold covering of the square S is known.

One can make sure that a 2-fold covering of the square S by eight circles ofthe same radius is realized, when each of four circles is replaced with twoones. The centres of introduced circles coincide with those of the initialones. Introduce constraint (2). With the same radius of circles as before,the 2-fold covering with r2 = 0.2 requires 10 circles. If r2 = 0.28, thenwe obtain a 2-fold covering of S by eight circles of the radius r1 = 0.380;with r2 = 0.2 we obtain that radii of eight circles that ensure a 2-fold cov-ering of S should equal r1 = 0.367. Therefore, the constraint imposed onthe distance between centres leads to the increase of either the quantityof circles, or their radii. The relief of the surface G is also essential; itstrongly affects the number of circles even with the increase of their radii.

According to results of numerical experiments, the proposed modelsand their solution methods are rather efficient.

References

1. Z. Drezner, H.W. Hamacher. Facility Location: Application and Theory,Springer-Verlag, NewYork (2004).

2. A.M. Fernandes, A.B. Utkin, A.V. Lavrov, R.M. Vilar. “Optimisation oflocation and number of lidar apparatuses for early forest fire detection inhilly terrain,” Fire Safety Journal, 41, 144–154 (2006).

3. Sh.I. Galiev, M.A. Karpova. “Optimization of Multiple Covering of aBounded Set with Circles,” Computational Mathematics and MathematicalPhysics, 50, No. 4, 721–732 (2010).

4. I.Kh. Sigal, A.P. Ivanova. Introduction to Discrete Programming: Modelsand Computational Algorithms, Fizmatlit, Moscow (2002).

5. http://www.fire-watch.de

6. http://www.lesdozor.ru

90

Stochastic subgradient barrier-multiplicative descentfor entropy optimization

Alexander Gasnikov1,2, Eugenia Gasnikova2

1 Premolab MIPT, Moscow, Russia;2 FAMC MIPT, Moscow, Russia;

Consider an entropy-linear programming problem in the following gen-eral form:

−

m∑k=1

xk ln(xk/e)→ max~x∈A∩R

m+

A : Λ(~x) = ~q − T~x = ~0l; F (~x) = ~d−G~x ≥ ~0w, l+ w ≤ m(1)

A family of dual barrier-multiplicative algorithms for finding a uniquesolution of (1) was suggested in [1]. This family contains Shelekhovskij,MART, GISM, Popkov’s algorithms and has the following form:

xnk = exp

−

l∑p=1

tpkλnp −

w∑q=1

gqkµnq

, k = 1, . . . ,m

λn+1p = λnp − γgλp

(qp −

m∑k=1

tpkxnk

), p = 1, . . . , l

µn+1q = µnq − αµnq gµq

(dq −

m∑k=1

gqkxnk

), q = 1, . . . , w

(2),

where γ, λ > 0 are sufficiently small constants,

gλp (·)lp=1, gµq (·)wq=1 ∈ C2

are monotone increasing functions, equal to zero on a null vector.A typical choice for those functions is

gλp (y) = − ln(1− y/qp), p = 1, . . . , l; gµq (y) = y/dq, q = 1, . . . , w.

Theorem 1. Assume that

1. ∃ ~z ≥ ~0m : ~q − T~z = ~0l, ~d−G~z ≥ ~0w,

2. Rows of matrices T = ‖tpk‖l,mp,k=1 and G = ‖gqk‖w,mq,k=1, if takentogether, are independent.

91

Then the iterative process (2) is globally convergent.

The second assumption is usually not fulfilled in practice. So we willpresent a new proposition.

Theorem 2. Assume that there ∃ ~z ≥ ~0m : ~q−T~z = ~0l, ~d−G~z ≥ ~0w.Then there exist sufficiently small γ, λ > 0, such that the iterative process(2) is globally convergent.

Iterative algorithm (2) is particularly effective, if T and G are sparsematrices or if l + w << m. Then the algorithm (2) will have complexityO(m+ l + w) or O(m · (l + w)) at the each step.

General formulation of (2) allows us to formulate also the stochasticversions of it. In practice, sometimes it is more computationally effec-tive to zero out via some probabilistic algorithm most of the gλp (·)lp=1,gµq (·)wq=1 components. This probabilistic algorithm will base on a cur-rent state of (2) and have a logarithmic complexity (see [4]). One shouldremember to scale the resulting vector in order to keep its norm closeto the gradient norm (see [5]). So we end up with sort of a stochasticsubgradient descent in a dual space (see [6], [7]).

The work was supported by RFBR 11-01-00494-a. The first author is par-

tially supported by Laboratory for Structural Methods of Data Analysis in l

Predictive Modeling, MIPT, RF government grant, ag. 11.G34.31.0073.

References

1. Gasnikova E.V.Dual multiplicative algorithms for an entropy-linear program-ming problem // Zh. Vychisl. Mat. Mat. Fiz. 2009. V. 49:3. P. 453-464.

2. Popkov Y.S. Macrosystems Theory and Its Applications. Springer Verlag,London, 1995.

3. Fang S.-C., Rajasekera J.R., Tsao H.-S.J. Entropy optimization and mathe-matical programming. Kluwer’s International Series, 1997.

4. Knuth D.E., Yao A.C. The complexity of nonuniform random number gen-eration // Algorithms and Complexity. N. Y.: Academic press, 1976. P.357-428.

5. Ermoliev Yu., Wets R.J.-B. Numerical Techniques for Stochastic Optimiza-tion (Springer Series in Computational Mathematics). Springer-Verlag, 1988.

6. http://www2.isye.gatech.edu/˜nemirovs/

7. http://www.core.ucl.ac.be/˜nesterov/

92

An Efficient Algorithm for Determining the LowerConvex Hull of a Finite Point Set in 3D

Dinh Thanh Giang1, Phan Thanh An2, Le Hong Trang3

1 CEMAT, Instituto Superior Tecnico, Av. Rovisco Pais 1049-001 Lisboa,

Portugal; [email protected] Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam;

[email protected] Department of Electrical Engineering, K.U.Leuven, Kasteelpark Arenberg 10,

3001 Leuven, Belgium; [email protected]

To date, the lower convex hull of a finite point set is determined fromthe entire convex hull of the set. There arises a question “Can we deter-mine the lower convex hull of a finite point set without rely on the entireconvex hull?”

In this paper, some properties of lower facets and lower convex hulls aregiven. Among the either, we show that the lower convex hull is wrappedby lower facets starting from an extreme edge of the lower convex hull.Then an efficient algorithm for determining the lower convex hull of afinite point set in 3D without the entire convex hull is presented. Theactual run times on the set of random points (in uniform distribution) ona paraboloid show that our algorithm runs significantly faster than theincremental convex algorithm implemented in [2] and some versions of thegift-wrapping algorithm (see [1] and [3]).

References

1. P. T. An and L. H. Trang. “An efficient convex hull algorithm for finite pointsets in 3D based on the Method of Orienting Curves,” Optimization, DOI:10.1080/02331934.2011.623163 (2011).

2. J. O’Rourke. Computational Geometry in C, Cambridge University Press,Second Edition (1998).

3. F. P. Preparata and M. I. Shamos. Computational Geometry - An Introduc-tion, Springer-Verlag, New York, Second Edition (1988).

93

LP projection algorithm and Newton method forsolving dual LP problems

Alexander I. Golikov1

1 Computing Center RAS, Moscow, Russia; [email protected]

Large-scale LP problems usually have more than one solution. Suchtechniques as the simplex methods, interior point methods make it pos-sible to obtain different solutions in the case of non-uniqueness. For ex-ample, the simplex method yields a solution belonging to a vertex ofpolyhedron. Some variants of the interior point method converge to asolution satisfying the strict complementary slackness condition.

LP projection method is close to the quadratic penalty function methodand to the modified Lagrangian function method. This method yields theexact projection of a given point on the solution set of primal LP problemas a result of the single unconstrained maximization of an auxiliary piece-wise quadratic concave function for any sufficiently large values of thepenalty parameter. A generalized Newton method with a stepsize chosenusing Armijo’s rule was used for unconstrained maximization. The proofof globally convergent finitely terminating generalized Newton method forpiecewise quadratic function was giving in [1], [2]. LP projection methodsolves LP problems with a very large (≈ 107) number of variables andmoderate (≈ 105) number of constraints [3], [4].

In a similar way, the exact projection of a given point on the so-lution set of the dual LP problem can be obtained by nonnegative con-strained maximization of auxiliary quadratic function for sufficiently largebut finite values of the penalty parameter [5]. Unfortunately the Newtonmethod can not be applied to nonnegativity constrained maximizationproblem directly but there exists a very simple way to overcome thisshortcoming.

Consider the primal linear program in the standard form

f∗ = minx∈X

c⊤x, X = x ∈ Rn : Ax = b, x ≥ 0n (P )

together with its dual

f∗ = maxu∈U

b⊤u, U = u ∈ Rm : A⊤u ≤ c, (D)

94

where A ∈ Rm×n, c ∈ Rn and b ∈ Rm are given, x is a primal variableand u is a dual variable, 0i denotes the i-dimensional zero vector.

Consider the problems of finding the least 2-norm projection u∗ of thepoint u on the solution set U∗:

1

2‖u∗ − u‖2 = min

u∈U∗

1

2‖u− u‖2, (1)

U∗ = u ∈ Rm : A⊤u ≤ c, b⊤u = f∗.Here, the Euclidian norm of vectors is used, and f∗ is an a priory unknownoptimal value of the objective function of the original LP problems (P )and (D).

Consider the following maximization problem on the positive orthantRn+ :

maxy∈Rn

+

S1(y, α, u), (2)

S1(y, α, u) = −c⊤y + u⊤Ay − 1

2‖αb−Ay‖2

Here the scalar α is fixed. A solution y(α) to problem (2) is closelyconnected with the solution u∗ to problem (1).

Theorem 1. [5] There exists α∗ such that for all α ≥ α∗ the uniqueleast 2-norm projection u∗ of a point u on U∗ is given by

u∗ = u+ αb−Ay(α),

where y(α) is a point maximizing S1(y, α, u) on Rn+.Theorem 1 makes it possible to replace problem (1), which contains an

a priori unknown value f∗, by problem (2), which involves the half-interval[α∗,+∞] instead of this value. This essentially simplifies the calculations.Note that the value α∗ may be negative. This occur when the projectionof the point u on the solution set U∗ coincides with the projection of thispoint on the feasible set U . The estimation of the threshold value α∗ isgiven in [5].

Theorem 2. [5] For all α > 0 and all u = u∗ ∈ U∗ an exact solutionto primal problem (P ) is given by u∗ = y(α)/α, where y(α) is a pointmaximizing S1(y, α, u∗) on Rn+.

Unfortunately the Newton method can not be applied to nonnegativ-ity constrained maximization problem (2) directly. By incorporating the

95

nonnegativity constraint y ≥ 0n into the objective function of (2) as apenalty term, we have unconstrained maximization problem

maxy∈Rn

−c⊤y + u⊤Ay − 1

2‖αb−Ay‖2 − τ

2‖(−y)+‖2,

where τ > 0 is a penalty parameter. In this case we obtained the optimalsolution y only in limit as τ → +∞. This property complicates the com-putation. There exists a very simple way to overcome this shortcoming.

Let a vector w ∈ Rm+n consists of two vectors w⊤ = [u⊤, v⊤], whereu ∈ Rm, v ∈ Rn. Consider the following LP problem

f∗ = maxw∈W

b⊤u, (D′)

W = u ∈ Rm, v ∈ Rn : A⊤u+ v = c, v ≥ 0n,which is equivalent to dual problem (D). The solution set of this problemis denoted byW∗ = [U∗×V∗]. For a given point w we find the least 2-normprojection w∗ on W∗ as a solution to following minimization problem

1

2‖u∗ − u‖2 +

1

2‖v∗ − v‖2 = min

w∈W∗

12‖u− u‖2 + 1

2‖v − v‖2,

W∗ = u ∈ Rm, v ∈ Rn : A⊤u+ v = c, v ≥ 0n, b⊤u = f∗.

Similarly to the approach which was used above we come to followingunconstrained maximization problem

maxy∈Rn

S2(y, γ, w), (3)

where S2(y, γ, w) = −c⊤y + u⊤Ay − 1

2‖γb−Ay‖2 − 1

2‖(v − y)+‖2.

The following Theorems hold [7].Theorem 3. There exists γ∗ such that for all γ ≥ γ∗ the unique least

2-norm projection w⊤∗ = [u⊤∗ , v

⊤∗ ] of a point w⊤ = [u⊤, v⊤] on W∗ is given

byu∗ = u+ γb−Ay(γ),v∗ = (v − y(γ))+,

where y(γ) is a solution to unconstrained maximization problem (3).

96

Theorem 4. For all γ > 0 and w = w∗ ∈ W∗ an exact solution toprimal problem (P ) is given by x∗ = y(γ)/γ, where y(γ) is a solution tounconstrained problem (3).

To solve the primal and dual LP problems simultaneously, one can usethe following iterative process:

ys+1 ∈ arg maxy∈Rn

−c⊤y + u⊤s Ay −1

2‖γb−Ay‖2 − 1

2‖(vs − y)+‖2, (4)

us+1 = us + γb−Ays+1, (5)

vs+1 = (vs − ys+1)+. (6)

Theorem 5. For all γ > 0 and for arbitrary starting point w0 theiterative process (4)–(6) converges to w∗ ∈W∗ in a finite number of iter-ations σ. The formula x∗ = yσ+1/γ gives an exact solution to the primalproblem (P ).

The goal function S2(y, γ, w) of unconstrained maximization problem(3) is piecewise quadratic concave function. Therefore one can defineits generalized Hessian which is m×m symmetric negative semi-definitematrix:

∂2

∂y2S2(y, γ, w) = −A⊤A−D(z).

Here, D(z) denotes the n× n diagonal matrix with diagonal elements zi,i = 1, 2, . . . , n. If (v − y)i > 0 then zi = 1, if (v − y)i ≤ 0 then zi = 0.Now for solving problem (3) one can use generalized Newton method.

There is an important difference between problem (2) and problem(3). In the first case we look for the projection of a given point u on U∗,in the second case we project a point W = [u, v] on the solution set W∗.Let u1∗ and u2∗ denote the projections of point u and [u, v] in the first andsecond cases, respectively. Then following inequality holds:

‖u1∗ − u‖ ≤ ‖u2∗ − u‖.

The comparison of LP projection methods (which were implementedin MATLAB) with some well-known commercial (CPLEX) and researchsoftware packages showed that they are competitive with the simplex andthe interior point methods [4].

Several parallel versions of the generalized Newton method for solvinglinear programs based on various data decomposition schemes of matrix

97

A (column, row, and cellular schemes) were implemented (see [6]). Theresulting parallel algorithms were successfully used to solve large-scale LPproblems (up to several dozens of millions of variables and several hun-dreds of thousands of constraints) for a relatively dense matrix A. Thecomputational experiments were performed on the cluster consisting oftwo-processor nodes based on 1.6 GHz Intel Itanium 2 processors con-nected by Myrinet 2000. For example, for LP problem with one millionvariables and 10000 constraints, the cellular scheme for 144 processorsof the cluster accelerated the computations approximately by a factor of50, and the computation time was 28 s. LP problem with two millionvariables and 200000 constraints was solved in about 40 min. on 80 pro-cessors. Another LP problem with 60 million variables and 4 thousandconstraints was solved by column scheme in 140 s. on 128 processors.

This research was supported by the Programs of Fundamental Research of

Russian Academy of Sciences P-15, P-18, by the Russian Foundation for Basic

Research (grant no.11-01-00786), and by the Leading Scientific Schools Grant

no.5264.2012.1

References

1. O.L. Mangasarian. ”A Newton Method for Linear Programming,” J. of Op-tim. Theory and Appl., 121, No.1, 1–18, (2004).

2. C. Kanzow, H. Qi, L. Qi,. ”On the Minimum Norm Solution of Linear Pro-grams,” J. of Optim. Theory and Appl., 116 No.2, 333–345, (2003).

3. A.I. Golikov, Yu.G. Evtushenko. ”Solution Method for Large-Scale LinearProgramming Problems,” Doklady Mathematics, 70, No.1, 615–619, (2004).

4. Yu.G. Evtushenko, A.I. Golikov, N. Mollaverdi. ”Augmented Lagrangianmethod for large-scale linear programming problems,” Optim. Methods andSoftware Apr-May; 7 Nos.4-5, 515–524 (2005).

5. A.I. Golikov, Yu.G. Evtushenko. ”Finding the Projection of a Given Point onthe Set of Solutions of a Linear Programming Problems,” Proceedings of theSteklov Institute of mathematics, Suppl.2, S68-S83, (2008).

6. Yu.G. Evtushenko, V.A. Garanzha, A.I. Golikov, H.M. Nguyen. ”Paral-lel Implementation of Generalized Newton Method for Solving Large-ScaleLP Problems,” in: Victor Malyshkin, ed. Parallel Computing Technologies.Lecture Notes in Computer Science 5698, Springer, Berlin, Heidelberg, NewYork, 2009, pp. 84–97

7. Y. Evtushenko, and A. Golikov. ”Linear Programming Projection Algo-rithms,” in: Wiley Encyclopedia of Operations Research and ManagementScience, 2011.

98

Many-Person Games With Convex Structure

Evgenij Golshtejn1

1 CEMI RAS, Moscow, Russia; [email protected]

Let Xi be a nonempty subset of a Euclidean space Ei, for 1 ≤ i ≤ k,k ≥ 2, and Xi the direct product of the subsets Xj for 1 ≤ j ≤ k, j 6= i,

X = Xi× Xi, E = E1× · · ·×Ek; and, consider real-valued functions fi isdefined on X , for 1 ≤ i ≤ k.

Introduce a k-person non-cooperative game Γ defined by the i-thplayer’s strategy set Xi and payoff function fi, for each 1 ≤ i ≤ k. Wesay that the game Γ satisfies condition C1 if X is a convex compact, andthe payoff function fi is continuous over X , concave and Lipschitz withrespect to xi ∈ Xi for any (fixed) xi ∈ Xi with the Lipschitz constant

independent on xi ∈ Xi, for any 1 ≤ i ≤ k.If condition C1 holds, the game Γ has a nonempty set X∗ of Nash

equilibrium points, and each payoff function fi is super-differentiable withrespect to xi ∈ Xi. Further, define a point-to-set mapping TΓ: X → 2E

as

TΓ(x) = t = (t1, . . . , tk) : − ti ∈ ∂xifi(x), 1 ≤ i ≤ k, x ∈ X,

where ∂xifi(x) is the super-differential of fi with respect to xi calculated

at the point x.Since the set X∗ of Nash equilibrium points of the game Γ coincides

with the set of solutions to the variational inequality

t ∈ T (x), 〈t, x′ − x〉 ≥ 0, ∀x′ ∈ X

if T = TΓ, then the methods solving variational inequalities can be usedto find the elements of the set X∗. In our previous paper [1], we de-scribed an efficient numerical algorithm to solve variational inequalities.Its convergence for T = TΓ is guaranteed if Γ satisfies condition C1 andthe mapping TΓ is monotone.

We say that a non-cooperative game Γ satisfying condition C1 hasa convex structure if the mapping TΓ is monotone. Any antagonisticgame with condition C1 valid has a convex structure; however, for non-antagonistic games it is not true in general.

99

A game Γ is said to satisfy condition C2 whenever condition C1 holdsand, apart from that, the i-th player’s payoff function fi is convex withrespect to xi ∈ Xi for any xi ∈ Xi, 1 ≤ i ≤ k, whereas the total payofff(x) =

∑ki=1 fi(x) of the game Γ is a concave function over x ∈ X .

Theorem 1. If the payoff functions fi of the game Γ can be repre-sented as fi(x) = f0

i (x) + ϕ(xi), x ∈ X, 1 ≤ i ≤ k, where the functionsf0i (x), x ∈ X, 1 ≤ i ≤ k, define a game satisfying condition C2, then thegame Γ has a convex structure.

Consider a finite non-cooperative game with k players, where playeri has ni strategies, and his payoff being determined by a k-dimensional

table Ai = (a(i)s1...sk); here a

(i)s1...sk -is the i-th player’s payoff when player

α chooses strategy sα, 1 ≤ α ≤ k. Having extended the players’ strategysets by allowing mixed strategies, we come to the game Γ, in which

Xi = xi = (xi1, . . . , xini) :

ni∑

j=1

xij = 1, xij ≥ 0,

fi(x) =∑

s1...sk

A(i)s1...skx1s1 × · · · × xksk ,

1 ≤ i ≤ k, x = (x1, . . . , xk).

Theorem 2. Let Γ be a finite non-cooperative game with mixed strate-gies defined by tables Ai, 1 ≤ i ≤ k. The game Γ has a convex struc-ture if and only if the tables Ai can be represented in the following formAi =

∑kj=1 Aij , 1 ≤ i ≤ k, where the entries of the k-dimensional table

Aij depend only upon the indices si and sj when i 6= j, but do not dependon the index si for i = j. Moreover, for any i 6= j, all entries of the tableAij +Aj i are zero.

Theorem 2 implies that the requirements to a game Γ listed in The-orem 1 guaranteeing that Γ has a convex structure, are not only suffi-cient but are also necessary if the game Γ belongs to the class of finitenon-cooperative many-person games with mixed strategies. Furthermore,for this class of games, the assumptions that the payoff functions f0

i

are concave with respect to xi and convex by xi, whereas the functionf0 =

∑ki=1 f

0i is concave with respect to x, turn out to be equivalent to

the following properties: the functions f0i are linear by xi on Ei and affine

with respect to xi over Xi, while f0 is affine on X .

100

References

1. E.G. Golshtein “A method for solving variational inequalities definedby monotone mappings,” Computational Mathematics and MathematicalPhysics, 42, No. 7, 921–930 (2002).

Proximal Analysis and Regularity of ViscositySolution to some Hamilton-Jacobi Equation

Vladimir Goncharov1, Fatima Pereira2

1 Universidade de Evora, CIMA-UE. Evora, Portugal; [email protected] Universidade de Evora, CIMA-UE. Evora, Portugal; [email protected]

Our talk is devoted to the Hamilton-Jacobi equation of a special form:

ρF (∇u (x))− 1 = 0, (1)

where F is a closed convex bounded subset of a Hilbert space (H, ‖ · ‖),containing the origin in its interior, and ρF (·) is the Minkowski functional(gauge function) associated to F ,

ρF (ξ) := inf λ > 0 : ξ ∈ λF .

This type equations arise, e.g., in geometric optics such as the famouseikonal equation

‖∇u(x)‖ − a = 0,

a > 0, in H = R3, or, in general, the elliptic equation

3∑

i=1

c2i u2xi

+2

c〈−→v ,∇u〉 − 1 = 0,

describing the propagation of a light wave from a point source placed atthe origin in anisotropic medium moving with a constant velocity −→v . Here

101

ci > 0 are the (constant) coefficients of refraction of light rays parallel tothe coordenate axes, and c means the speed of the light in a vacuum.

We are interested in regularity properties of the viscosity solution u(·)to the equation (1) on an open region Ω ⊂ H with the boundary datau|∂Ω = θ where the function θ(·) satisfies a kind of slope condition withrespect to F . It is well known that in this case the viscosity solution takesthe form

u (x) = infy∈CρF 0 (x− y) + θ (y) , (2)

where C := H \ Ω, and it can be interpreted also through the valuefunction in a minimum-time problem with a dynamics associated to thepolar set F 0.

Since, in general, viscosity solution is not smooth everywhere out ofC, our goal is to study the (Frechet) differentiability of u(·) just near C(or near the boundary ∂Ω).

Notice that the differentiability of the function (2) strongly relateswith the existence, uniqueness and regularity of minimizers in the respec-tive (mathematical programming) problem. In the case of compact C itfollows, for instance, from the representation of the Clarke’s subdifferen-tial of a marginal function through Radon measures supported on the setof minimizers (see [1, Section 2.8]). So, denoting by πF,θC (x), x 6∈ C, theset of minimizers in (2) we find first rather general conditions guarantee-

ing that πF,θC (x) is a singleton (Lipschitz) continuous near a given pointx0 ∈ ∂C. These conditions involve a ballance between the (proximal)subgradients of the restriction θ |C and the normals to F .

Then, under the same (local) hypotheses assuming, in addition, thesmoothness either of θ |C or of F we prove that the function u(·) is(Frechet) differentiable in (x0 + δB) \ C for some δ > 0 (with eventu-ally Holder continuous gradient ∇u(·)).

To achieve our goals we apply the mixed technique of Convex Analysiswith some tools of Proximal Calculus (see [2, Chapter 1]) such as thefuzzy sum rule and the Ekeland’s variational principle. Observe that thebasic local hypothesis naturally generalizes one of the geometric conditionsobtained in [3, 4] for the well-posedness of the minimum-time problem

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with a constant convex dynamics (the case θ ≡ 0). Some illustrativeexamples in finite dimensional spaces are given.

The authors were supported by Fundacao para Ciencia e Tecnologia (FCT),

the Portuguese institutions COMPETE, QREN and the European Regional

Development Fund (FEDER) (project PTDC/MAT/111809/2009 ”Variational

Analysis: Theory and Applications”).

References

1. F.H. Clarke. Optimization and Nonsmooth Analysis, Willey-Interscience,New York (1983).

2. F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, P.R. Wolensky. Nonsmooth Analysisand Control Theory, Springer, New York (1998).

3. V.V. Goncharov, F.F. Pereira. ”Neighbourhood retractions of nonconvex setsin a Hilbert space via sublinear functionals”, J. Convex Anal., 18, 1–36 (2011).

4. V.V. Goncharov, F.F. Pereira. ”Geometric conditions for regularity in a time-minimum problem with constant dynamics”, J. Convex Anal., 19 (2012), toappear.

Multimethod’s algorithm for parametric identificationof nonlinear dynamic systems

Aleksander Gornov1

1 ISDCT SB RAS, Irkutsk, Russia; [email protected]

The problem of parametrical identification of dynamic systems con-tinues to remain an actual problem of mathematical modeling. The mainmissions arising at the solution of task of parametrical identification ofdynamic systems, problems of nonconvexity of optimized functionals andpossible stiffness of studied system of the differential equations continueto remain.

The report examines the multimethod’s algorithms are oriented to thesolution of applied problems arising in the numerical study of dynamicalmodels. As basis of work the previous researches of author published in[1-6].

103

For numerical integration of dynamic systems the set of algorithmsincluding both options of the elementary methods of type Runge-Kutta,and the adaptive program technologies developed by experts DOPRI5and DOPRI8 for nonstiff systems, RADAU and RADAU5 for stiff is real-ized. At digitization of system will be applied a grid with the non-uniformstep, generated by applied algorithms individually to each of the Cauchyproblem. Accuracy of integration is supervised by the specialized algo-rithm, capable to estimate both an error of digitization of continuoussystem and an error of algorithms of integration.

One of the main installations applied by development of new algo-rithms, refusal from traditional in global optimization, but low-constructivefor this class of problems of hypotheses was considered. In particular, dueto lack of the possibility of obtaining analytical formulas for the deriva-tives of the algorithms required for adaptive technology assessment of thedifference gradients. On the other hand, calculation of the functionals de-fined on solutions of the Cauchy problem, does almost impossible use ofreliable interval methods and doesn’t allow to receive aprioristic estimatesof the Lipschitz constant.

To solve the nonconvex optimization problems formulated in the iden-tification system, developed a standardized set of optimization algorithms,including both the modification of known methods - multistart, simulatedannealing, genetic search, differential evolution, MSBH (Monotonic Se-quence Basin-Hopping), and new algorithms by taking into account thespecific characteristics of this class of problems.

Among the new algorithms may be mentioned the ”algorithm of parabo-las,” based on the nonlocal one-dimensional search algorithmwith parabolicconvex triples and refinement of descent, ”the tunneling algorithm”, scan-ning the search space by means of spline approximation to a randomdirection, ”pass-algorithm” that implements the search in the neighbor-hood extreme points of the new shutter with a concave profile, and themodification of the Powell algorithm, which allows jump out from localextrema.

For specification of the found local extrema some local algorithms,allowing to specify the decision without use of the gradients which are in-evitably containing integration noise are realized. The set of the describedbase algorithms formed a basis for creation of multimethod computingschemes.

The realized interface system allows to make trial calculations in a

104

dialogue mode and to design for each task the unique algorithm, capableto make active the best lines of each base algorithm. In particular, inmany tasks the combination of algorithms of differential evolution, localdescent on the Cauchy method and algorithm of MSBH perfectly proved.

The proposed algorithms are standardized, equipped with algorithmicparameters and realized in language C in the form of a scientific prototypeof the program complex functioning under control of OS Windows. Theefficiency of the algorithms studied on a number of test problems. Withapplication of the offered algorithms a number of applied tasks from ecol-ogy and nanophysics area is solved. The maximum size of system of thedifferential equations for which problems of parametrical identification aresolved, makes 550 phase variables.


(project no. 12-01-00193, no. 10-01-00595).

References

1. A.Yu. Gornov “The implementation of the random multistart for optimalcontrol problems,” Proc. of Lyapunov Conference, Irkutsk. p. 31 (2003).

2. A.Yu. Gornov, T.S. Zarodnuk “Optimal Control Problem: Heuristic Algo-rithm for Global Minimum,” Proc. of the Second International Conferenceon Optimization and Control, Ulanbaatar, Mongolia. pp. 27–28 (2007).

3. T.S. Zarodnuk, A.Yu. Gornov “Search technology of the global extremum inoptimal control problems,” Modern Technology. Systems analysis. Simula-tion, vol. 4. pp. 70–76 (2008).

4. A.Yu. Gornov The computational technologies for solving optimal controlproblems, Nauka, Novosibirsk (2009).

5. A.Yu. Gornov, T.S. Zarodnuk “The method of “curved search” of the globalextremum in optimal control problems,” Modern Technology. Systems anal-ysis. Simulation, vol. 3. pp. 19–26 (2009).

6. A.Yu. Gornov “Global extremum search algorithms in the optimal controlproblems,” Proc. of International Conference “Optimization and applica-tions” (OPTIMA-2009), Montenegro, Petrovac. p. 36 (2009).

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Optimization Problems in Astrodynamics

Anna D. Guerman1

1CAST - Centre for Aerospace Science and Technologies, University of Beira

Interior, Covilha, PORTUGAL; [email protected]

Planning of a space mission has to take into account several limitationson the available resources and restrictions on possible solutions, so largespectrum of optimization problems naturally arise in the context. Sincethe beginning of the Space Era such problems as optimal reorientationmaneuver or minimum time trajectory transfer have been in the focus ofattention of researchers; their successful solution made possible a numberof breathtaking missions in early 60th and still are the source of inspira-tion for many space scientists and engineers. Nowadays, the equipmentand energy available on-board offer one much more options for in-orbitmaneuvers, and many new optimization methods have been developed tomake the best of new technology.

We present a review of a number of problems related to analysis ofmodern space missions, from attitude maneuvers to orbital control.

Blaschke Convergence Theorem for G-type ConvexSets in Metric Spaces

Nguyen Ngoc Hai1, Phan Thanh An2

1Department of Mathematics, International University, Vietnam National

University, Ho Chi Minh City, Vietnam; [email protected] of Mathematics, Vietnam Academy of Science and Technology, 18

Hoang Quoc Viet Road, 10307 Hanoi, Vietnam [email protected]

The Blaschke convergence theorem is a classical theorem in convexanalysis. It says that a uniformly bounded infinite collection of closed

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convex sets in a finite dimensional space contains a sequence which con-verges to a nonempty compact convex set. There are generalizations ofthis theorem for infinite dimensional normed linear spaces (see [7]) or forsets that are not convex in the usual sense such as star-shaped sets (see[5]). In [4], we dealt with analytic properties of geodesic convex sets ina simple polygon. Among the others, Blaschke’s convergence theorem forgeodesic convex sets was presented. The aim of this paper is to developsome basic ideas in [4] to get a generalization of Blaschke’s convergencetheorem in metric spaces. First, we will introduce the notion of G-typeconvexity. The usual convexity in linear spaces and geodesic convexityin uniquely geodesic spaces are special cases of G-type convexity. Someproperties of convex sets are still true for G-type convex sets (Propositions1 and 3 (ii)). We then get a generalization of Blaschke’s convergence the-orem for metric spaces (Theorem 7). In particular, in a proper uniquelygeodesic space, every uniformly bounded sequence of nonempty geodesi-cally convex sets contains a subsequence which converges to a nonemptycompact geodesically convex set (Corollary 8). Theorem 7 also reduces tothe classical Blaschke convergence theorem when X is a finite dimensionalnormed linear space and G(x, y) is the line segment joining x and y.

D e f i n i t i o n 1. A generalized geodesic space is a metric space (X, d)together with a set-valued mapping G : X × X → 2X that satisfy thefollowing conditions:

(A1) G(x, y) 6= ∅ for all x, y ∈ X ;(A2) If (xn) and (yn) are sequences in X , xn → x and yn → y, then

dH(G(xn, yn), G(x, y)

)→ 0 as n→∞

where the metric on a normed linear space (X, ‖ · ‖) is the usual metricd(x, y) = ‖x − y‖ and dH is the Hausdorff metric. We denote this gen-eralized geodesic space by (X, d,G). Then, any normed linear space is ageneralized geodesic space with G(x, y) = [x, y], any nonempty convex setin a normed linear space is a generalized geodesic space.

D e f i n i t i o n 2. Let X be a set and G : X×X → 2X a set-valuedmapping that takes on nonempty values. A set A ⊂ X is called G-typeconvex if G(x, y) ⊂ A whenever x, y ∈ A.

If X is a linear space and G(x, y) = [x, y], then G-type convexity coincides

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with the usual convexity.

Proposition 1. The intersection of an arbitrary family of G-type convexsets is G-type convex.

Proposition 2. If A is a nonempty G-type convex set in a generalizedgeodesic space (X, d,G) then (A, d,G) is a generalized geodesic space, too.

Proposition 3. Let (X, d,G) be a generalized geodesic space.(i) If xn, yn ∈ X, xn → x, yn → y and z ∈ G(x, y) then there is

zn ∈ G(xn, yn), n = 1, 2, . . . such that zn → z as n→∞.(ii) If A is a G-type convex subset of X, then so is its closure A.

Corollary 4. (See [6], p. 68) Let X be a proper uniquely geodesic spaceand let A be a geodesically convex subset of X. Then its closure A isgeodesically convex.

Consider now the collection G(X) = G(x, y) : x, y ∈ X.

Theorem 5. Let (X, d,G) be a compact generalized geodesic space.(i) For any sequence

(G(xn, yn)

), there exists a subsequence

(G(xnj

, ynj))

such that dH(G(xnj

, ynj), G(x, y)

)→ 0 for some x, y ∈ X. So if G(u, v)

is closed for all u, v ∈ X then(G(X), dH

)is a compact metric space.

(ii) If A is a nonempty compact subset of X and dH(G(xn, yn), A

)→

0, then A = G(x, y) for some x, y ∈ X.

Proposition 6. Let(An)be a sequence of nonempty G-type convex sub-

sets of a generalized geodesic space (X, d,G) and let A be a nonemptyclosed subset of X. If dH(An, A)→ 0, then A is also G-type convex.

Let us denote by GC(X) the collection of all nonempty G-type convexclosed subsets of a generalized geodesic space (X, d,G). A sequence ofsets in X is said to be uniformly bounded if there exists some ball in Xthat contains every member of the sequence (see [10], p. 96). We can nowformulate our main result of this paper.

Theorem 7. Let (X, d,G) be a generalized geodesic space.(i) If X is compact, then so is

(GC(X), dH

).

(ii) If X is proper, then every uniformly bounded sequence of nonemptyG-type convex sets in X contains a subsequence which converges to some

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nonempty compact G-type convex set in X.

A particular case of Theorem 7 (i) whenX is a simple polygon P on theplane and G(x, y) is the shortest segment connecting x and y was provedin [4]. We now have a direct and important consequence of Theorem 7.

Corollary 8. If X is a proper uniquely geodesic space then every uni-formly bounded sequence of nonempty geodesically convex subsets of Xcontains a subsequence which converges to some nonempty compact geodesi-cally convex subset in X.

In fact, in Theorem 7 and Corollary 8 X may be any metric spaceprovided that there is a compact set Y containing all members of thesequence

(An). The proof of Theorem 7 (ii) remains valid for this case.

Proposition 9. Let(An)be a sequence of nonempty G-type convex sub-

sets of a generalized geodesic space (X, d,G). If there exists a compactset that contains all members of

(An), then there exists a subsequence of(

An)which converges to some nonempty compact G-type convex set in

X.

Corollary 10. (See [7]) If C is a nonempty compact convex set in anormed linear space, then every sequence of nonempty compact convexsubsets of C contains a subsequence which converges to some nonemptycompact convex set in C.

Finally we can obtain the classical Blaschke convergence theorem forconvex sets in IRn, which is a special case of Theorem 7 (ii).

Corollary 11. (See [10], p. 97) Every uniformly bounded sequence ofnonempty compact convex sets in IRn contains a subsequence which con-verges to some nonempty compact convex set in IRn.

References

1. C. D. Aliprantis, O. Burkinshaw Principles of Real Analysis, 3rd ed, Aca-demic Press, 1998.

2. J. P. Aubin Applied Abstract Analysis, John Wiley & Sons, New York, 1977.

3. M. R. Bridson and A. Haefliger Metric spaces of non-positive curvature,Springer, 1999.

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4. N. N. Hai and P. T. An “Blaschke-type theorem and separation of disjointclosed geodesic convex sets,” Journal of Optimization Theory and Applica-tions, 15 (3), 541–551 (2011).

5. T. Hirose “On the convergence theorem for star-shaped sets in En,” Proc.Japan Acad., 41 (3), 209–211 (1965).

6. A. Papadopoulos Metric Spaces, Convexity and Nonpositive Curvature, Eu-ropean Mathematical Society, 2005.

7. G. B. Price “On the completeness of a certain metric space with an applica-tion to Blaschke’s Selection Theorem,” Bull. Amer. Math. Soc., 46, 278–280(1940).

8. G. T. Toussaint “Computing geodesic properties inside a simple polygon,”Revue D’Intelligence Artificielle, 3, 9–42 (1989).

9. F. A. Valentine Convex Sets, McGraw-Hill, New York, 1964.

10. R. Webster Convexity, Oxford University Press, 1994.

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Pathwise optimal control of diffusion type processes

Niyaz Ismagilov1, Farit Nasyrov2

1 Ufa State Aviation Technical University, Ufa, Russia;

[email protected] Ufa State Aviation Technical University, Ufa, Russia; [email protected]

Introduction

In the work, we consider a stochastic optimal control problem of diffu-sion type processes with pathwise cost functional, that is, the problemof finding a control function such that it minimizes cost for every singletrajectory of state variable.

We introduce new method of solving problems of pathwise cost mini-mization. The main idea of the method is that original stochastic controlproblem can be reduced to deterministic control problem. Solution to thelatter gives pathwise optimal solution to the original problem.

Problem Statement

Let (Ω, F, P ) be a probability space. Consider a stochastic differentialequation defined on this space describing dynamics of some system

dxt = b(t, xt, ut) dt+ σ(t, xt, ut) ∗ dWt, x(0) = x0. (1)

In the above equation t ∈ [0, T ] ⊂ R; state variable function xt is areal-valued stochastic process; control function ut is a stochastic processtaking values in V ⊂ R with paths in class of piecewise continuous func-tions; b(t, x, u) and σ(t, x, u) : [0, T ]×R× V → R are known determinedfunctions, which we will call drift coefficient and diffusion coefficient re-spectively; Wt(ω) is a standard Wiener process and its differential ∗ dWt

in right-hand side of (1) is interpreted in the sense of Stratonovich integral.We introduce a cost functional to measure performance of control

J =

∫ T

0

f0(t, xt, ut)dt. (2)

The problem is to find control that minimizes the functional (2) subjectto dynamics equation (1).

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Previous work on optimal control of diffusion processes has mainlybeen concerned with problem of finding control that minimizes “mean”value of cost

J = E

∫ T

0

f0(t, xt, ut)dt→ min,

where E denotes expectation [1],[2]. In contrast, stated above problemis concerned with minimization of functional (2), which represents costfor every single path of state variable. Hence pathwise optimality is thedistinguishing feature of this work.

In the work we consider two particular cases of problem (1), (2):

A Control function only affects on drift coefficient, i.e. instead of (1)dynamics of system is

dxt = b(t, xt, ut) dt+ σ(t, xt) ∗ dWt, x(0) = x0. (3)

The problem is to minimize functional (2) subject to (3).

B Control function consist of two components u1t and u2t which affecton drift coefficient and diffusion coefficient respectively, and the dif-fusion coefficient is linear w.r.t. u2t , i.e. instead of (1) dynamics ofsystem is

dxt = b(t, xt, u1t ) dt+ σ(t, xt)u

2t ∗ dWt, x(0) = x0. (4)

The problem is to minimize functional

J =

∫ T

0

f0(t, xt, u1t , u

2t )dt. (5)

Solution to the problems

Solution to problem A. It’s known ([3]) that xt is a solution to theequation (3) if and only if it can be represented in the form

xt = ϕ∗(t,Wt + yt), (6)

where ϕ∗(t, v) is known function determined by diffusion coefficient andnot depending on control or state variable, and yt is a solution to thefollowing Cauchy problem

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dytdt

=b(t, ϕ∗(t,Wt + yt), ut)− (ϕ∗)′t(t,Wt + yt)

σ(t, ϕ∗(t,Wt + yt)), (7)

y(0) = y0 : ϕ∗(0,W0 + y0) = x0, (8)

here (ϕ∗)′t(t,Wt + yt) is partial derivative of ϕ∗(t, v) w.r.t. t evaluated at

point (t,Wt + yt). Hence (3) is equivalent to (6)–(8).As function ϕ∗(t, v) doesn’t depend on control we conclude that all

the affect of control on state variable xt = ϕ∗(t,Wt + yt) is focused onyt. The latter allows to consider constrain (7) instead of (3) leading us tothe new problem with state variable y, same control function, dynamicsequation (7) and cost functional

J =

∫ T

0

f0(t, ϕ∗(t,Wt + yt), ut)dt, (9)

that is (6) substituted in (2).Thus, problem (3), (2) is equivalent to problem (7)–(9) which has an

advantage of having non-stochastic differential equation as dynamics ofsystem. That is, despite having random functions (these areWt(ω), yt(ω)and ut(ω)) in the right-hand side of equation (7), for almost all ω ∈ Ωthe equation is an ordinary differential equation of the form dyt/dt =f(t, yt, ut). Hence the problem (7)–(9) is deterministic optimal controlproblem and can be solved using classical deterministic methods as e.g.maximum principle.

Solution to problem B. The idea of solution to problem B is similarto the one presented for problem A, but with a slightly different result.Solution to equation (4) is presented as xt = ϕ∗(t,Wtu

2t + yt) (compare

to (6)), where ϕ∗(t, v) is again known function determined by diffusioncoefficient σ and not depending of control or state variable, and yt is asolution to the following Cauchy problem

dytdt

=b(t, ϕ∗(t,Wtu

2t + yt), u

1t )− (ϕ∗)′t(t,Wtu

2t + yt)

σ(t, ϕ∗(t,Wtu2t + yt))u2t−Wtνt,

du2tdt

= νt.

(10)

y(0) = y0, u2(0) = u20. (11)

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In the same way as in solution to problem A, a new problem withdynamics given by (10), (11) and cost functional

J =

∫ T

0

f0(t, ϕ∗(t,Wtu2t + yt), u

1t , u

2t )dt, (12)

is equivalent to problem (4),(5).The difference to the case of problem A is that in reduced problem

(10)–(12) instead of single equation of dynamics there is a system of twodifferential equations (10) with generalized derivatives, and u2t is treatedas a state variable among with yt. Control functions are u1t and νt, thelatter is an impulse control function, i.e. represented as sum of absolutelycontinuous function and linear combination of Dirac delta functions

νt = νct +∑

i

hiδ(t− τi).

Thus problem B reduces to deterministic optimal impulse controlproblem (10)–(12) which can be solved using corresponding methods.

Conclusion

The method introduced in work provides capability of solving pathwisestochastic optimal control problems. Reduction of initial stochastic prob-lem to the new deterministic problem lies at the heart of the method. It isshown that problems with control affecting only drift coefficient reduce toclassical deterministic optimal control problems and problems with con-trol affecting both drift and diffusion coefficients reduce to deterministicoptimal impulse control problems.

Reduced problems can be solved using deterministic methods as e.g.maximum principle. The main difficulty in applying deterministic meth-ods to reduced problems is that coefficients of the equations (7), (10) mayhave unbounded variation in t, as a result be not differentiable. Further-more, optimal control found by this method not always turns out to benon-anticipating.

References

1. N.V. Krylov. Controlled Diffusion Processes, Nauka, Moscow (1977) (in Rus-sian).

114

2. I.I. Gichman, A.V. Skorochod Controlled stochastic processes, NaukovaDumka, Kiev (1977) (in Russian).

3. F.S. Nasyrov. Local times, symmetric integrals and stochastic analysis, Fiz-matlit, Moscow (2011) (in Russian).

On accuracy of the regularization method ofconstrained ill-posed quadratic minimization

problems

M. Jacimovic1, I. Krnic2

1 Montenegrin Academy of Scineces and Arts, Podgorica, Montenegro;

[email protected] Department of Mathematics, University of Montengro, Podgorica,

Montenegro

Let H and F be real Hilbert spaces, A : H 7→ F - bounded linearoperator from H to F , f ∈ F - a fixed element, and U ⊆ H- a closedconvex set. We will deal with the minimization problem

J(u) =1

2‖Au− f‖2 → inf, u ∈ U. (1)

which is is equivalent to the variational inequality

find u ∈ U : 〈A∗Au −A∗f, v − u〉 ≥ 0, ∀v ∈ U. (2)

In case of U = H , it is equivalent to the operator equation

A∗Au = A∗f. (3)

Problems of the type (1) with infinite-dimensional spaces H and Fare usually present in linear optimal control problems. We will shortlypresent one example of such problems (see [5]).

Given T > 0, find u = u(·) ∈ U = u ∈ L2[0, T ] :∫ T0 u2(t)dt ≤ r2,

such that

J(u) =

∫ l

0

|x(T, s;u)− f0(s)|2ds+

115

∫ l

0

|x′t(T, s;u)− f1(s)|2ds,

be minimal. Here f0(·) ∈ H10 [0, 1], f1(·) ∈ L2[0, l] are given functions and

x = x(t, s;u) is the solution of the equation

∂2x(t, s)

∂t2=∂2x(t, s)

∂s2+ u(t),

x(t, 0) = x(t, l) = 0, 0 < t < T ;

x(0, s) = 0,∂x(t, 0)

∂t= 0, 0 < s < l.

Problem (1) in literature (see [4]) is regularly studied under assumptionthat instead of the exact operator A and instead of the element f oneactually deals with their approximations Aη ∈ L(H,F ), fδ ∈ F, suchthat

‖A−Aη‖ ≤ η, ‖f − fδ‖ ≤ δ, ‖Aη‖2 ≤ a, (4)

where η > 0, δ > 0 are small positive real numbers and a > 0.

In general case, problems (1) and (3) are ill-posed. This fact generatesthe necessity to apply some methods of regularization [1], [3], [5], [4], thatwill produce good approximate solutions of the problems.

Usually the bounds of the accuracy of the regularization methods forsolving ill-posed problems (1) and (3) were obtained for the problems thatsatisfy so-called source conditions or sourcewise representable conditions.In this paper, we will consider the power source condition (see [1],[2], [3]and [4])

u∞ = |A|ph∗, where h∗ ∈ H, |A|p = (A∗A)p2 , p > 0, (5)

and projected power source conditions (see [3]),

u∗ = πU (|A|ph∗), h∗ ∈ H, p > 0. (6)

that were used widely for obtaining the estimates of the convergence rateof regularization methods for solving linear operator equations. Here, u∞and u∗ are the solutions of (3) and (1) with minimal norms (so-callednormal solutions). The source condition seems quite natural for problem(3) if we have in mind that u∞ ∈ R(A∗), where R(A∗) is the range of the

116

operator A and R(A∗) its closure in norm of the space H . It means thatthe solution u∞ is densely surrounded by the elements from R(A∗). In ageneral case, the normal solution u∗ of (1) does not belong to R(A∗), butu∗ ∈ πU (R(A∗)).

As a regularized approximate solution of problem (1), we will considerthe solution u = uα of the variatiaonl inequality

〈gα(A∗ηAη)u −Aηfδ, v − u〉 ≥ 0, ∀v ∈ U, (7)

where gα : [0, a] 7→ R, (α > 0), are Borel measurable functions satisfyingthe following conditions:

1− tgα(t) ≥ 0, t ∈ [0, a], (8)

1

t+ βα≤ gα(t) ≤

1

βα, t ∈ [0, a], β > 0, (9)

(∃p0 > 0)∀p ∈ [0, p0] sup0≤t≤a

tp(1 − tgα(t)) ≤ γαp, (10)

where γ = γ(p0). Here, a is the constant from (4). Number p0 is calledqualification of the family gα : α > 0.

Note that the functions gα(t) = (t+α)−1 and gα(t) =∑m−1j=0

αj

(t+α)j =

t−1(1− (1 + t)−m) (that defines Tikhonov methods of regularization andits iterated variant) satisfy these conditions. Similar methods of regu-larization of opeartor equations were used widely in literature (see, forexample [1], [4]).

The next theorem contains the conditions of the convergence of theregularized solution uα to normal solution u∗ of (1).

Theorem. Suppose conditions (4) and (8)-(10) are satisfied.(a) If the parameter α in (7) is chosen such that α = α(η, δ)→ 0 and

η+δ2

α → 0 as η, δ → 0, then uα → u∗ as η, δ → 0.

(b) If one of the conditions (5) or (6), and gα(t) = 1/(t+α) is satisfied,and

α = α(η, δ) = d(η + δ)2/(p+2), d = const,

then‖uα − u∗‖ = O(η + δ)p/(p+2), 0 ≤ p ≤ 2p0 − 1,

‖Aη(uα − u∗)‖ = O(η + δ).

117

References

1. A.B. Bakushinsky, M. Yu. Kokurin. Iterative methods for approximate solu-tions of inverse problems, Springer, Dordrecht (2004).

2. M. Jacimovic, I. Krnic. “On some classes of regularization methods for min-imization problem of quadratic functional on a halfspaces,” Hokkaido Math.Journal, 28, 57–69 (1999).

3. A. Neubauer. “Tikhonov-regularization of ill-posed linear operator equationson closed convex sets,“ Journal of Appr. Theory, 53, 304–320 (1988).

4. G.M. Vainikko, A.Yu. Veretennikov. Iterative Procedures in Ill-Posed Prob-lems, (in Russian), Nauka, Moscow (1986).

5. F.P. Vasil’ev. Methods of optimization, (in Russian), Factorial, Moscow(2003).

Cubes Lattice’s properties investigation andpossibilities of its application in Combinatorial

Optimization

Ruben V. Khachaturov1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; rv [email protected]

A new type of lattices — Lattice of Cubes (Cubes Lattice) [1, 2] isunder the investigation. It is shown, that the number of all sub-cubes ofthe M -dimensional cube is equal to 3M , and the set of all such sub-cubes(with a corresponding choice of union and intersection operations) formsa lattice named a Lattice of Cubes [1, 2]. Algorithms of construction ofsuch lattices are described, results of work of these algorithms for variousdimensions of lattices are illustrated (fig. 1, 2). The total number ofelements of such lattice (including the empty set) is equal to 3M+1, whereM is dimension of the lattice. It is proved, that the Lattice of Cubes isa lattice with relative supplement, that allows to use effective algorithms[3, 4, 5] for solving on it the problems of minimization and maximization ofsupermodular functions. Concrete examples of such functions are given.

118

Optimization algorithms and possibilities of setting and solving a newclass of problems on the Cubes Lattices are discussed.

Lemma.The number |Kr| of all r-dimensional subcubes (0 ≤ r ≤ m)in the cube Cm is equal to

|Kr| = 2m−rCrm = 2m−r m!

r!(m− r)! .

Theorem 1.The total number of subcubes of all dimensions in anm-dimensional cube Cm is 3m.

The proofs of these Lemma and Theorem can be found in [1, 2, 5].Let’s define the union and intersection operations for the Cubes Lat-

tices.D e f i n i t i o n 1. If C1, C2 ∈ K — elements of Cubes Lattice K

and C1 = (ω11 ;ω

12), C2 = (ω2

1 ;ω22), ω

11 ⊂ ω1

2 , ω21 ⊂ ω2

2 , then

C1 ∪ C2 = C = (ω11 ∩ ω2

1; ω12 ∪ ω2

2),

C1 ∩ C2 =

C(ω1

1 ∪ ω21 ; ω

12 ∩ ω2

2), if (ω11 ∪ ω2

1) ⊂ (ω12 ∩ ω2

2),

∅, if (ω11 ∪ ω2

1) 6⊂ (ω12 ∩ ω2

2).

Theorem 2.The Cubes Lattice K is a lattice with relative supplement.Proof. To prove this theorem it is enough to show, that for any

element of the lattice C1 ∈ K there will be at least one element C2 ∈ Ksuch that C1 ∪ C2 = (0; I), C1 ∩ C2 = ∅.

Indeed, let C1 = (ω11 ;ω

12), where ω

11 ⊂ ω1

2 , and C2 = (ω21 ;ω

22), where

ω21 ⊂ ω2

2 . Let’s show, that the element of the lattice C2, at which ω21 = 0

and ω22 = I \ ω1

2, is relative supplement to the element C1.It can be proved by direct check, using the above-stated definitions of

union and intersection operations for elements of the Cubes Lattice K, asfollows:

C1 ∪C2 = (ω11 ;ω

12) ∪ (0; I\ω1

2) = (0; I).

C1 ∩ C2 = (ω11 ;ω

12) ∩ (0; I\ω1

2) = (ω11 ; 0) = ∅.

The theorem is proved.Note: the relative supplement to each element may not be unique.Union operation allows to introduce a partial order between the ele-

ments C ∈ K. The following important definition is based on this opera-tion:

119

D e f i n i t i o n 2. C1 = (ω11 ;ω

12) ⊂ C2 = (ω2

1 ;ω22) if C1 ∪ C2 = C2,

i.e. when ω11 ⊃ ω2

1 and ω12 ⊂ ω2

2 .Two algorithms for constructing diagrams of a set of all C ∈ K (i.e.

— Lattice of Cubes), based on this definition, have been worked out.First algorithm. At a zero level of the Lattice of Cubes diagram

there is always only one element — an empty subset (∅). At the firstlevel we shall arrange all 2m sub-cubes containing only one element, i.e.cubes C0 = (ω1;ω2), where ω1 = ω2. Then, using the union operation, weshall construct elements of the second level. At this level we shall arrangeonly cubes C1, and so on. This process of constructing the Cubes Lat-tice is similar to the modified algorithm of successive calculations [3, 4]with modifications corresponding to the union operation. In the processof constructing the next levels of the cubes diagram, we construct only apart of elements, as it is done in the algorithm of successive calculations,rather than construct simultaneously all elements of each level. This al-lows to effectively apply the corresponding rejection rules while solvingthe optimization problems [3, 4, 5].

Note: The total number of elements of m-dimensional Cubes Lattice,including an empty set ∅, is 3m + 1.

Fig. 1. The Lattice of Cubes diagram for m = 3

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Figure 1 shows the Lattice of Cubes (Cubes Lattice) constructed usingthe first algorithm for m = 3.

Second algorithm. In the beginning the minimal set ω1 = (0, . . . , 0)is fixed and all sets ω2 ⊃ ω1 are looking through under the order ofincrease in binary system of calculation from ω2 = ω1 up to ω2 = I.During this we construct the corresponding lattice elements and the ribsconnecting each constructed element with its neighborhood. Then weadd 1 to ω1 in binary system, and look through all ω2 ⊃ ω1 again fromω2 = ω1 up to ω2 = I. This is repeated until ω1 becomes equal to I.Thus, we obtain all elements of the Cubes Lattice, passed through all 2m

of independent ways, that allows to solve quickly some kinds of problemsof discrete optimization. In addition, this algorithm allows to arrangeelements of the lattice on a plane in accordance with 2m basis vectors.

Figure 2 shows the Lattice of Cubes (Cubes Lattice) constructed usingthe second algorithm for m = 4.

Fig. 2. The Lattice of Cubes diagram for m = 4

References

1. V.R. Khachaturov, R.V. Khachaturov. “Lattice of Cubes,” Journal of Com-puter and Systems Sciences International (Izvestiya RAN), 47, No. 1, 40–46(2008).

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2. V.R. Khachaturov, R.V. Khachaturov. “Cubes Lattices and supermodularoptimization,” in: Works of the third international conference devoted tothe 85-anniversary of the correspondent member of the Russian Academy ofSciences, professor L.D. Kudryavtsev, publishing house of MFTI, Moscow,2008, pp. 248–257.

3. V.R. Khachaturov. Mathematical Methods of Regional Programming,Economic-mathematical library, Nauka, Moscow (1989).

4. V.R. Khachaturov, V.E. Veselovskiy, A.V. Zlotov, et al. Combinatorial Meth-ods and Algorithms for Solving the Discrete Optimization Problems of theBig Dimension, Nauka, Moscow (2000).

5. V.R. Khachaturov, R.V. Khachaturov, R.V. Khachaturov. SupermodularProgramming on Finite Lattices, Communications on applied mathematics,Dorodnicyn Computing Centre of RAS, Moscow (2009).

General Theory of Optimization on Finite Lattices

Vladimir R. Khachaturov1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; rv [email protected]

The report consists of the Introduction and five main sections. In Sec-tion 2 the basic concepts, definitions and optimization problems settingare given. In Section 3 we describe the problems of minimization of su-permodular functions on the different types of lattices: Boolean lattices,lattices with relative supplements (division lattices, lattices of vector sub-spaces of finite-dimensional vector space, geometrical lattices), latticesequal to Cartesian product of chains. The previously obtained theoreticalresults, on the basis of which the problems of minimization of supermodu-lar functions on these lattices have been solved, are described. It’s noted,that these results have been extended to the distributive lattices.

Section 4 is devoted to the elaboration of the new basic propositions ofthe theory of maximization of supermodular functions on Boolean lattices(they were worked out only for the problems of minimization before) andestablishing of the relation between the global minimum and maximum ofsupermodular functions for the main types of lattices. In Section 5 a newtype of lattices — Lattice of Cubes, is defined and described [1, 2]. The

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problems of minimization and maximization of supermodular functionsare also considered on it.

In Section 6 we describe a general approach to the optimization onlattices with use of atomic lattices. We propose to map the atomic latticeinto the corresponding Boolean lattice and then perform the optimizationon this more ample Boolean lattice. If the properties of supermodularityof the function, defined on the atomic lattice, are obeyed on the Booleanlattice, then for the optimization it is possible to use all the theoreticalresults from Sections 3–5. Therefore, the elaboration of different effec-tive algorithms of representation of high-dimensional Hyper-Cubes is ofbig significance. We describe the original combinatorial algorithms of au-tomated representation of high-dimensional Hyper-Cubes (Booleans) ona plane in the form of different projections and diagrams, keeping theproperties of Boolean as a partially ordered set of its vertexes. This givesus the ample opportunities for construction of various schemes of look-ing through the elements of atomic lattices and for visualization of theoptimization process. The examples of various types of applied problemsthat have been solved using the elaborated optimization methods [3] aredescribed. Although the majority of these problems are NP-hard, solu-tion of a great amount of applied tasks demonstrated the high practicaleffectiveness of the elaborated methods and algorithms [3, 4]. (Analogywith the simplex-method can be seen.)

Thus, the obtained theoretical results and a great amount of optimiza-tion problems for lattices with concrete types of supermodular functionsallow to consider the methods and algorithms for solving the problemsof optimization of supermodular functions on lattices as a new field ofmathematical programming — supermodular programming.

References

1. V.R. Khachaturov, R.V. Khachaturov. “Lattice of Cubes,” Journal of Com-puter and Systems Sciences International (Izvestiya RAN), 47, No. 1, 40–46(2008).

2. V.R. Khachaturov, R.V. Khachaturov. “Cubes Lattices and supermodularoptimization,” in: Works of the third international conference devoted tothe 85-anniversary of the correspondent member of the Russian Academy ofSciences, professor L.D. Kudryavtsev, publishing house of MFTI, Moscow,2008, pp. 248–257.

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3. V.R. Khachaturov. Mathematical Methods of Regional Programming,Economic-mathematical library, Nauka, Moscow (1989).

4. V.R. Khachaturov, V.E. Veselovskiy, A.V. Zlotov, et al. Combinatorial Meth-ods and Algorithms for Solving the Discrete Optimization Problems of theBig Dimension, Nauka, Moscow (2000).

Hyperplane Covering Problems. Complexity andApproximation Issues

Michael Khachay1, Maria Poberii2

1 Institute of Mathematics and Mechanics, UB of RAS, Ekaterinburg, Russia;

[email protected] Institute of Mathematics and Mechanics, UB of RAS, Ekaterinburg, Russia;

maschas [email protected]

Settings of geometric covering problem and related problems are usualin various operations research domains [1-3]: optimal facility location the-ory, cluster analysis, pattern recognition, etc. Mathematically, family ofsuch problems can be participated into two classes.

The first one contains special cases of well-known abstract Set Coverproblem. The main general feature of all these problems is the finitenessof the initial family of subsets, for which it is required to find a subfam-ily (or just prove its existence) covering some target set and satisfyinggiven optimality conditions. There are many papers studying problemsfrom this class (see survey at [4]). The classical papers [5-7] seem to bethe most important among them. First two papers contain intractabilityproof of Set Cover problem and two main design patterns for construct-ing approximation algorithms for this problem. The last paper proves theoptimality of these patterns, unless P = NP.

The second class consists of problems without the mentioned abovefiniteness constraint. Usually, the initial family of subsets is given hereimplicitly in terms of some geometric property characterizing its elements.For instance, for a given set it is required to find a minimal cardinalitycover by straight lines, circles of a given radii, etc.

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In the paper, a series of hyperplane covering problems for given fi-nite sets in finite-dimensional vector spaces of fixed dimension d > 1 isconsidered. The first element of this family (for d = 2), also known asPoint Covering on the plane (2PC) problem was studied by N.Megiddoand A.Tamir [8] who proved its intractability in the strong sense.

We extend this result on to the case of appropriate fixed dimension-ality d > 1 and prove that all these problems are Max-SNP-hard andconsequently have no PTAS, unless P = NP.

P r o b l e m 1. ‘Point covering by lines on the plane’ (2PC). A finitesubset P = p1, . . . , pn ⊂ Z

2 and natural number B are given. Is thereexists a cover C of P by straight lines such that |C| ≤ B?

Obviously, in the particular case when the set P is in the generalposition, i.e. each triple of its points does not belong to the same line, the2PC problem has a trivial solution (’Yes’ whether B ≥ ⌈|P |/2⌉ and ’No’otherwise), which can be found in a polynomial time. But in the generalcase this problem is intractable.

Theorem 1 [8]. The 2PC problem is NP-complete in the strong sense.

Let us consider the more general problem settings.P r o b l e m 2. ‘Hyperplane covering in d-dimensional space’ (dPC).

For a fixed d > 1, a finite subset P = p1, . . . , pn ⊂ Zd and natural

number B are given. Is there exists a cover C of P by hyperplanes suchthat |C| ≤ B?

P r o b l e m 3. ‘Minimal hyperplane covering in d-dimensional space’(Min-dPC). Let a finite subset P = p1, . . . , pn ⊂ Z

d be given. It isrequired to find a minimum cardinality partition J1, . . . , JL of a set Nn =1, . . . , n such that for each i ∈ NL there is a hyperplane Hi and

pj ∈ P : j ∈ Ji ⊂ Hi.

We start with construction of polynomial-time reduction of (d− 1)PCto dPC problem. Let an instance of (d − 1)PC be given by subset P =p1, . . . , pn ⊂ N

d−1M and B ∈ N. We use a natural isomorphic embedding

of (d− 1)-dimensional into d-dimensional vector space:

x ∈ Rd−1 7→ [x, 0] ∈ R

d.

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Map any point pi ∈ P into couple of points in Zd by the formula

p2i−1 = [pi,−wi], p2i = [pi, wi],

where

wi = (K + 2)i−1 and K =⌈(d− 1)

d−12 (M − 1)d−1

⌉.

Such a way, we construct the subset P ⊂ Zd and the setting (P , B) of the

dPC problem.It is evident, that any hyperplane cover of P induces the equivalent

cover (with the same number of hyperplanes) of P in Rd. The converse

statement should be proved.Denote by π0 the hyperplane [x, 0] : x ∈ R

d−1. Let Prπ0 Q be anorthogonal projection of the subset Q ⊂ R

d onto π0.

Lemma 1. Let subsets Q ⊂ P and Q ⊂ P be related by Q = Prπ0 Qand the following inequalities be valid

|Q| ≥ d+ 1,

dimaffQ ≤ d− 1.

Then dim affQ ≤ d− 2.Lemma 2. Let Π = π1, . . . , πt be a hyperplane cover of subset P .

The subset P also has a hyperplane cover Π such that |Π| ≤ t.Lemma 3. The described above reduction (d − 1)PC to dPC can be

done in polynomial time of Length((d− 1)PC).

On the basis of these lemmas we can prove the followingTheorem 2. For an arbitrary fixed d > 1, the dPC problem is NP-

complete (and the Min-dPC problem is NP-hard) in the strong sense.

Now we show that the supposed above (d − 1)PC to dPC reductioncan be reformulated as L-reduction [9] from Min-(d − 1)PC to Min-dPCproblem.

D e f i n i t i o n 1. Let sets I and S, set-valued map F : I → 2S

and some target function c :⋃I∈I

F (I) → R+ be given. The quadrupleA = (I, S, F, c), where each I ∈ I is mapped to optimization problem

minc(s) : s ∈ F (I),

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is called a combinatorial minimization problem.W.o.l.g., any I ∈ I is called an instance of the problem A and its

optimum value is denoted by OPT (I).

D e f i n i t i o n 2. Consider problems A and B of combinatorialminimization. It is called, that there is an L-reduction from A into B,if there are two LSPACE-computable functions R and S and positiveconstants α and β such that the following conditions are valid:

1. for each instance I of the problem A, R(I) is an instance of B and

OPT (R(I)) ≤ αOPT (I);

2. for each feasible solution z of R(I), S(z) is a feasible solution of Isuch that

cA(S(z))−OPT (I) ≤ β(cB(z)−OPT (R(I))),

where cA, cB are target functions of A and B correspondingly.

Theorem 3. For each fixed d > 2, there is an L-reduction of Min-(d− 1)PC to Min-dPC problem.

Taking into account the following known resultTheorem 4 [11]. Min-2PC problem is Max-SNP-hard.one can formulate the lastTheorem 5. For each fixed d > 1, the Min-dPC problem is Max-

SNP-hard.

Consequently, Min-dPC problem has no polynomial-time approxima-tion schema (PTAS) for each fixed d > 1, unless P = NP.


(projects no. 10-01-00273 and 10-07-00134) and Ural Branch of RAS (projects

no. 12-P-1-1016 and 12-C-1-1017/1).

References

1. P.K. Agarwal, C.M. Procopiuc. “Exact and approximation algorithms forclustering”, Algorithmica, No. 33, 201–206. (2002).

127

2. S. Langerman, P. Morin. “Covering things with things ”, Discrete Computat.Geom., 717–729, (2005).

3. M. Khachai. “Computational complexity of recognition learning proceduresin the class of piecewise-linear committee decision rules”, Automation andRemote Control, 71, No. 3, 528–539, (2010).

4. V. Vazirany. Approximation algorithms, Springer (2001).

5. D. Johnson. “Approximation algorithms for combinatorial problems”, Journalof Computer and System Sciences, 9, No. 3, 256–278, (1974).

6. L. Lovasz. “On the ratio of integer and fractional covers”, Discrete Mathe-matics. No. 13, 383–390, (1975).

7. U. Feige. “A Threshold of lnn for Approximating Set Cover”, Journal of theACM, 45, No. 4, 634–652, (1998).

8. N. Megiddo, A. Tamir. “On the complexity of locating linear facilities in theplane”, Operations research letters. 1, No. 5, 194–197, (1982).

9. C. Papadimitriou, M. Yannakakis. “Optimization, approximation, and com-plexity classes”, J. Comput. System Sci., 43, No. 3, 425–440, (1991).

10. C. Papadimitriou. Computational Complexity, Addison-Wesley, (1995).

11. M. Khachai, M.Poberii. “Computational complexity of combinatorial prob-lems related to piecewise linear committee pattern recognition learning pro-cedures”, Pattern Recognition and Image Analysis. 22, No. 2, 278–290,(2012).

Leontief’s model as a boundary value problem inoptimal control

Elena Khoroshilova1

1 Lomonosov Moscow State University, Moscow, Russia;

[email protected]

1. Statement of problem. The optimal control problem with freeright end on a fixed interval is considered in this paper. The dynamics ofthe process is described by a system of ordinary differential equations

d

dtx(t) = D(t)x(t) +B(t)y(t), t0 ≤ t ≤ t1, x(t0) = x0 = 0, (1)

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with the trajectories1 x(·) ∈ PC1([t0, t1],Rn), and the controls y(·) ∈ Y .

Y = y(·) ∈ PC([t0, t1],Rn)| yi(t) ∈ [y−i , y+i ], i = 1, n, (2)

D(·), B(·) ∈ C([t0, t1],Rn×n). Let the right ends x1 = x(t1), y1 = y(t1) ofthe trajectories and controls satisfy the constraints

x1 = A1x1 + y1, y1 ≥ 0, (3)

A1 ∈ Rn×n. Complementing the system (1) with a control y(·) ∈ Y and

solving it, we find the trajectory x(·). When control changes within theset Y the right-hand ends x1 of trajectories describe the attainable set,on which the following objective function is defined

ϕ(x1, y1) = ϕ1(x1) + ϕ2(y1), (4)

where ϕ1(x1) and ϕ2(y1) are convex and differentiable in the variables x1and y1, respectively.

We need to determine the optimal control y∗(·) ∈ Y and the corre-sponding trajectory x∗(·) ∈ PC1([t0, t1],R

n), subject to the system (1).At the same time their right ends y∗1 and x∗1 have to minimize the objectivefunction (4) under constraints (2)–(3).

It is assumed that a solution exists, but not unique. Similar formula-tions of the problem, but without the terminal control, were examined in[1],[2]. In case ϕ(x1, y1) ≡ ϕ(y1), the optimization problem is similar tothe input-output economic model [3].

2. Lagrangian, dual problem and boundary-value problem.Consider the Lagrangian for the optimization problem

L(p1, ψ(·);x(·), y(·)) = ϕ1(x1) + ϕ2(y1)+

+〈p1, (I −A1)x1 − y1〉+∫ t1

t0

〈ψ(t), D(t)x(t) +B(t)y(t) − d

dtx(t)〉dt,

defined for all x(·) ∈ PC1([t0, t1],Rn), x1 ∈ R

n, y(·) ∈ Y , y1 ≥ 0, p1 ∈ Rn,

ψ(·) ∈ PC1([t0, t1],Rn)′ – the dual space.

1Here PC1([t0, t1],Rn) is the class of continuous vector-valued functions: [t0, t1] →Rn with piecewise continuous derivatives; PC([t0, t1],Rn) is the class of piecewise

continuous vector-valued functions: [t0, t1] → Rn.

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The point (p∗1, ψ∗(·); x∗(·), y∗(·)) is called a saddle point of Lagrange

function if for all x(·) ∈ PC1([t0, t1],Rn), x1 ∈ R

n, y(·) ∈ Y , y1 ≥ 0,p1 ∈ R

n, ψ(·) ∈ PC1([t0, t1],Rn)′ the following system holds

L(p1, ψ(·);x∗(·), y∗(·)) ≤ L(p∗1, ψ∗(·);x∗(·), y∗(·)) ≤ L(p∗1, ψ∗(·);x(·), y(·)).(5)

Here (x∗(·), y∗(·)) and (p∗1, ψ∗(·)) are called direct and dual variables.

According to the Kuhn-Tucker theorem, there exist p∗1 and ψ∗(·), suchthat if the pair (x∗(·), y∗(·)) is a solution, then (p∗1, ψ

∗(·); x∗(·), y∗(·))is a saddle point of the Lagrangian. Converse is also true: if the pair(x∗(·), y∗(·)) satisfies the saddle point system (5), then it is a solution tothe original problem.

Transforming the Lagrangian, as well as the saddle point system toconjugate form, and using the right-hand inequality of the system, weobtain the dual problem. Combining the direct and dual problems, wecome to a boundary value problem

d

dtx∗(t) = D(t)x∗(t) +B(t)y∗(t), x∗(t0) = x0, y

∗(·) ∈ Y, t0 ≤ t ≤ t1,

x∗1 = A1x∗1 + y∗1 , y∗1 ≥ 0,

d

dtψ∗(t) +DT (t)ψ∗(t) = 0, ψ∗

1 = ∇ϕ1(x∗1) + (I −AT1 )p∗1,

ϕ2(y1)− ϕ2(y∗1)− 〈y1 − y∗1 , p∗1〉+

∫ t1

t0

〈BT (t)ψ∗(t), y(t) − y∗(t)〉dt ≥ 0.

3. Method of solution. The method of simple iteration is thesimplest of the known numerical methods. However, in this case we aredealing with a saddle problem, for which the simple iteration method,generally speaking, not converge. Therefore, to solve the problem, we usean extra-proximal approach [4]–[5]:

1) prediction half-step

d

dtxk(t) = D(t)xk(t) +B(t)yk(t), xk0 = x0,

pk1 = pk1 + α((I −A1)xk1 − yk1 ),

d

dtψk(t) +DT (t)ψk(t) = 0, ψk1 = ∇ϕ1(x

k1) + (I −AT1 )pk1 ,

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(yk1 , yk(·)) = argmin1

2|y1 − yk1 |2 + α〈∇ϕ2(y

k1 )− pk1 , y1 − yk1 〉+

+1

2

∫ t1

t0

|y(t)−yk(t)|2dt+α∫ t1

t0

〈BT (t)ψk(t), y(t)−yk(t)〉dt | y1 ≥ 0, y(·) ∈ Y ;

2) basic half-step

d

dtxk(t) = D(t)xk(t) +B(t)yk(t), xk(t0) = x0,

pk+11 = pk1 + α((I −A1)x

k1 − yk1 ),

d

dtψk(t) +DT (t)ψk(t) = 0, ψk1 = ∇ϕ1(x

k1) + (I −AT1 )pk1 ,

(yk+11 , yk+1(·)) = argmin1

2|y1 − yk1 |2 + α〈∇ϕ2(y

k1 )− pk1 , y1 − yk1 〉+

+1

2

∫ t1

t0

|y(t)−yk(t)|2dt+α∫ t1

t0

〈BT (t)ψk(t), y(t)−yk(t)〉dt | y1 ≥ 0, y(·) ∈ Y .(6)

In the first half-step of each iteration we find the ”forward” vectors pk1 ,yk1 , y

k(·) (those that should be at the next step in the simple iterationmethod). Then in the second half-step we use them to find the directionof future movement on the (k+1)-th iteration. From the current positionin the k-th iteration, we take a step in that direction, and calculate pk+1

1 ,yk+11 and yk+1(·). Thus, when we choose the direction of movement, thendo not take into account the current gradient, but the gradient of the nextstep.

The following theorem on the convergence of the method was proved.Theorem 1. Let the set of optimal trajectories x∗(·) of the prob-

lem (1)–(4) be not empty and belong to the subspace PC1([t0, t1],Rn).

If ϕ1(x1) and ϕ2(y1) are convex and differentiable functions with gradi-ents satisfying a Lipschitz condition with constants L1, L2, D(t), B(t) arecontinuous matrices, Y is a set of the form (2). Then the sequence

|yk1 − y∗1 |2Rn + |pk1 − p∗1|2Rn + ‖yk(·)− y∗(·)‖2Ln2,

generated by method (6) with the choice of parameter α from the con-dition 0 < α < α0, where α0 = min( 1√

2L2, 1√

2L22+B

2maxC2

, 1Bmax

√C1

),

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Bmax, C1, C2 – some calculated constants, decreases monotonically. More-over, every weakly converging subsequence of controls yki(·) convergesweakly to the optimal control y∗(·), and the corresponding subsequence oftrajectories xki(·) converges to the optimal trajectory x∗(·) in the uni-form norm Cn[t0, t1].

If the sequence of controls yk(·) has at least one strong limit as k →+∞ then the process xk1 , yk1 , pk1 ;xk(·), yk(·), ψk(·) converges to a solutionof the problem x∗1, y

∗1 , p

∗1;x

∗(·), y∗(·), ψ∗(·), with respect to the variablesy(·), y1, p1 – monotonically.

4. Conclusions. The optimal control problem with free right endand linear differential equations constraints is considered. The right-handends of controls and trajectories generate a finite-dimensional Cartesianproduct, on which a minimum of the objective function is defined underconstraints such as input-output model of Leontief.

To solve this problem we suggest an iterative method of extra-proximaltype, consisting in the construction of the functional sequences of trajecto-ries and controls. We have proved that the sequences of controls, trajecto-ries, conjugate trajectories, as well as the sequences in finite-dimensionalspaces of primal and dual variables converge monotonically in norm tosolution of the problem.


(project no. 12-01-00783) and by the Program ”Leading Scientific Schools”

(NSh-5264.2012.1).

References

1. A.S. Antipin. “Boundary-value games in optimal control problems”, Opti-mization methods and their applications. XV Baikal International School-Seminar. Irkutsk, Plenary reports, 1, 21–29 (2011).

2. E.V. Khoroshilova. “The optimal control problem with the optimization of thetwo ends of the time interval”, Optimization methods and their applications.XV Baikal International School-Seminar. Irkutsk, Optimal control, 3, 150–155 (2011).

3. A.V. Lotov. Introduction to Economics and Mathematical Modeling, Nauka,Moscow (1984).

4. A.S. Antipin. “Iterative methods of predictive type for computing fixed pointsof extremal mappings”, Izvestia Vysshykh Uchebnykh Zavedeny. Matem-atika. No. 11(402), 17–27 (1995).

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5. A.S. Antipin, E.V. Khoroshilova. “On methods of extragradient type forsolving an optimal control problem with linear constraints”, Izvestia IGU.Matematika. 3, No. 3, 2–20 (2010). Online: http://isu.ru/izvestia

Approximation to minimum committee problem forsystem of linear inequalities in IR3

Konstantin Kobylkin1

1 Institute of Mathematics and Mechanics, Ural Branch of RAS, Ekaterinburg,


An NP -hard minimum committee problem for infeasible three-dimen-sional system of homogeneous linear inequalities (MC(3)) is considered.Well known minimum covering problem for finite set on the plane usingstraight lines is reducible to MC(3) [1]. A useful approximation to MC(3)is studied in the form of minimum committee problem for infeasible system

(cj , h) > bj , j = 1, . . . ,m, cj , h ∈ IR2, bj ∈ IR, rankci, cj = 2, i 6= j. (1)

D e f i n i t i o n 1. [2] A finite subset K of IR2 is called a committeeof a system (1) iff for each j ∈ Nm = 1, . . . ,m there exists a subsetK(j) ⊂ K with |K(j)| > |K|/2 whose elements satisfy jth inequality of(1). A committee of (1) having the least cardinality is called minimumcommittee.

D e f i n i t i o n 2. Let i, j ∈ Nm, i 6= j. We call ith inequality of (1)inessential with respect to its jth inequality (relative to system (1)) iff ithinequality belongs to each maximum feasible subsystem of (1) containingjth inequality.

Denote by J (j,∆bj) the system obtained from (1) by excluding in-equality (cj , h) > bj and adding inequality (cj , h) > bj+∆bj instead (withthe same number j), where ∆bj ∈ IR. Let ρ1(i, j) be exact lower bound of∆bj such that ith inequality is inessential with respect to jth inequalityrelative to the system J (j,∆bj). Also let ρ2(i, j) be exact lower boundof ∆bj such that there exists infeasible subsystem of three inequalities ofJ (j,∆bj) which contains inequalities (ci, h) > bi and (cj , h) > bj +∆bj .

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Theorem. If minρ1(i, j), ρ2(i, j) ≤ 0 for every distinct i, j ∈ Nm,then minimim committee cardinality for system (1) is the same as thatfor homogeneous system

(cj , h) > 0, j ∈ Nm. (2)

MC problem for system (2) is known to be O(m) hard [2]. As aconsequence MC problem for (1) is polynomially solvable under theoremconditions which can also be checked in polynomial time.

The authors were supported by Presidium of Ural Branch of RAS (projects

no. 12-P-1-1016 and 12-S-1-1017/1), and by Russian Foundation for Basic Re-

search (projects no. 10-01-00273 and 10-07-00134).

References

1. M.Yu. Khachay, M.I. Pobery. “Complexity and approximability of committeepolyhedral separability of sets in general position,” Informatica, 20, No. 11,217–234 (2009).

2. Vl.D. Mazurov. Committee method for optimization and classification prob-lems, Nauka, Moscow (1990).

Metric for the total tardiness minimization problem

Pavel Korenev1,2, Alexander Lazarev1,2,3

1 Institute of Control Sciences of the Russian Academy of Sciences, Russia;2 Moscow State University, Moscow, Russia;

3 National Research University Higher School of Economics, Moscow, Russia;


Introduction

Suppose that we have a set N = 1, 2, . . . , n of n jobs to be processed ona single machine. Preemptions are not allowed. The machine is available

134

since time t0 = 0 and can handle only one job at a time. Job j ∈ N isavailable for processing since its release date rj ≥ 0, its processing requiresprocessing time pj ≥ 0 time units and should ideally be completed beforeits due date dj . We will call an instance the set of given parameters:release dates, processing times, and due dates. We will use superscriptsto distinguish parameters belonging to different instances. Note that aninstance A = rA1 , . . . , rAn , pA1 , . . . , pAn , dA1 , . . . , dAn can be considered as avector in 3n-dimensional space.

Let Sj(π) and Cj(π) be the starting and the completion time of job j ∈N in schedule π, respectively. We will consider only early schedules, i.e.,if π = j1, . . . , jn, then Sj1 = max0, rj1, Sjk = maxrjk , Cjk−1

, k =2, 3, . . . , n, and Cj(π) = Sj(π) + pj , j ∈ N . Thus an early schedule isuniquely determined by a permutation of the jobs of set N . Then letTj(π) = max0, Cj(π)− dj be a tardiness of job j in schedule π.

The objective is to find an optimal schedule π which minimizes thetotal tardiness, i.e., objective function is F (π) =

∑j∈N

Tj(π). The problem

is denoted by 1|rj |∑Tj.

In the paper we propose a new approach for the total tardiness mini-mization problem. The approach is to construct a polynomially solvableinstance B and apply its solution to the given instance A. To evaluatethe error of the solution we construct a metric for the considered problem.For the problem 1|rj |

∑Tj we propose a metric ρ(A,B).

ρ(A,B) = n ·maxj∈N|rAj − rBj |+ n ·

∑

j∈N|pAj − pBj |+

∑

j∈N|dAj − dBj |.

This function can be considered as a metric for the problem and boundsdifference between optimal values of objective functions of instances Aand B.

Metrical approach

Theorem 1. The function

ρ(A,B) = n ·maxj∈N|rAj − rBj |+ n ·

∑

j∈N|pAj − pBj |+

∑

j∈N|dAj − dBj |.

satisfies the metric axioms.

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Theorem 2. Let πA and πB be an optimal schedules for instances Aand B, respectively. Moreover, let πB be an approximate schedule, subjectto

∑

j∈NTBj (πB)−

∑

j∈NTBj (πB) ≤ δ.

Then

∑

j∈NTAj (πB)−

∑

j∈NTAj (πA) ≤ 2ρ(A,B) + δ.

The idea of the metrical approach is to find the least distanced inthe metric from the given instance A polynomially solvable instance B.Then, by applying known polynomial algorithm to the instance B, oneobtains a schedule πB which can be used as an approximate solutionfor instance A with error no greater than 2ρ(A,B). One can also useapproximate solution for the instance B with an absolute error δ as anapproximate solution for instance A, in this case the error is not greaterthat 2ρ(A,B) + δ.

Thereby, the problem 1|rj |∑Tj is reduced to the function ρ(A,B)

minimization problem .Let us search for the instance B in the polynomially solvable class

defined by the system of linear inequalities

A · RB + B · PB + C ·DB ≤ H,where RB = (rB1 , . . . , r

Bn )

T , PB = (pB1 , . . . , pBn )

T , DB = (dB1 , . . . , dBn )

T ,pBj ≥ 0, rBj ≥ 0, j ∈ N , T is transposition symbol, A,B, C – m × nmatrices, and H – a column of m elements.

Then the problem of finding the least distanced from A instance of thegiven polynomially solvable class can be formulated as follows

minimize f = n · (yr − xr) + n ·∑

j∈N(ypj − xpj ) +

∑

j∈N(ydj − xdj ),

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subject to

xr ≤ rAj − rBj ≤ yr,xpj ≤ pAj − pBj ≤ ypj ,xdj ≤ dAj − dBj ≤ yd,rBj ≥ 0, pBj ≥ 0, j ∈ N,

A · RB + B · PB + C ·DB ≤ H.

It is the problem of the linear programming, with 7n + 2 variables:rBj , p

Bj , d

Bj , x

pj , y

pj , x

dj , y

dj , x

r, yr, j = 1, . . . , n.However, it is not necessary to use algorithms of the linear program-

ming, if there are less complicated ways.The metrical approach can be applied to other scheduling problems,

if a metric function with required properties is constructed.Lemma 1. Consider the scheduling problem with following objective

function

F (π) =∑

j∈Nφj(π, r1, . . . , rn, p1, . . . , pn, dj).

Then the function

ρ(A,B) =∑

j∈N

∑

i∈N(Rji|rAj − rBj |+ Pji|pAj − pBj |) +

∑

j∈NDj|dAj − dBj |,

where Rji ≥ |∂φi

∂rj|, Pji ≥ |∂φi

∂pj|, Dj ≥ |∂φj

∂dj|, can be used as a metric for the

problem, and the metrical approach can be applied to find an approximatesolution of the problem.

Computational experiments

We used three polynomially solvable classes in computational experi-ments. These classes are PR : pj = p, rj = r, j ∈ N, PD : pj =p, dj = d, j ∈ N, RD : rj = r, dj = d, j ∈ N. In the optimal sched-ules for these classes jobs are processed in the increasing order the freeparameter.

Lemma 2. Minimum of the metric function ρ(A,B), where B ∈PR,PD,RD can be found in O(n) operations.

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Table 5: Average experimental error in percentage of the theoretical errorn PR PD RD4 2,5% 4,6% 20,8%5 2,6% 4,8% 23,1%6 2,6% 4,6% 24,6%7 2,6% 4,7% 26%8 2,5% 4,6% 27%9 2,4% 4,7% 27,9%10 2,4% 4,6% 28,6%

To evaluate approximate solutions for both cases we have run compu-tational experiments. For each value of n and each of used polynomiallysolvable classes 10000 instances were generated. Experiments were per-formed for n = 4, 5, . . . , 10. For each instance, processing times pj weregenerated randomly in the interval [1, 100], due dates dj were generated inthe interval [pj,

∑j∈N

pj], and release dates rj were generated in the interval

[0, dj − pj ]. We used proposed approach to find an approximate solutionwith value of objective function Fa for each instance, and branch & boundalgorithm to find an optimal solution with value of objective function Fo.After we estimated experimental error δ = Fa − Fo in percentage of thetheoretical error, which is doubled value of function ρ(A,B) .

All obtained distributions are bell-shaped. Obtained average errorsare shown in Table 1. In the PR-case experimental errors averages near2, 5% of the theoretical, in PD-case average error is near 4, 5% and inRD-case error grows from 20% to 30% with increasing of n

Conclusion

In the paper we have proposed the new approach to the total tardinessminimization problem. The approach is based on search for the poly-nomially solvable instance which has a minimal distance in the metricfrom the original instance. In further research we are going to improvethe approach by constructing new metrics and finding new polynomiallysolvable cases of scheduling problems.

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(projects no. 11-08-01321 and 11-08-13121).

References

1. P. Baptiste. “Scheduling equal-length jobs on identical parallel machines,”Discret. Appl. Math., No. 103, 21–32 (2000).

2. J. Du, J.Y.-T. Leung. “Minimizing total tardiness on one machine is NP-hard,” Math. Oper. Res., No. 15(3), 483–495 (1990).

3. R.L. Graham, E.L. Lawler, J.K. Lenstra, and A.H.G. Rinnoy Kan. “Opti-mization and approximation in deterministic sequencing and scheduling: asurvey,”Ann. Discret. Math., No. 5, 287–326 (1979).

4. E.L. Lawler. “A Pseudopolynomial Algorithm for Sequencing Jobs to Mini-mize Total Tardiness,”Ann. Discret. Math., No. 1, 331–342 (1977).

5. E.L. Lawler. “A fully polynomial approximation scheme for the total tardinessproblem,”Oper. Res. Lett., No. 1, 207–208 (1982).

6. A.A. Lazarev, A.G. Kvaratskheliya. “Metrics in Scheduling Problems,”Dokl.Math., No. 81, 497–499 (2010).

7. A.A. Lazarev, F. Werner. “Algorithms for Special Single Machine TotalTardiness Problem and an Application to the Even-Odd Partition Prob-lem,”Math. and Comp. Model., No. 49, 2078–2089 (2009).

New CQ-free optimality criterion for convex SIPproblems with polyhedral index sets

Olga Kostyukova1, Tatiana Tchemisova2

1 Institute of Mathematics, National Academy of Sciences of Belarus, Minsk,

Belarus; [email protected] University of Aveiro, Aveiro, Portugal; [email protected]

Consider a convex Semi-Infinite Programming (SIP) problem in theform

(P ) : minx∈Rn

c(x), (1)

s.t. f(x, t) ≤ 0 ∀ t ∈ T = t ∈ Rs : hTk t ≤ ∆hk, k ∈ K, (2)

139

where the objective function c(x), x ∈ Rn, is convex, the constraint func-

tion f(x, t), x ∈ Rn, t ∈ T, is linear w.r.t. x; hk ∈ R

s, ∆hk ∈ T, k ∈K, |K| <∞. Notice that the index set T in (P ) is a convex polyhedron.

Let X be the feasible set of problem (P ): X = x ∈ Rn : f(x, t) ≤

0, ∀t ∈ T . Suppose that f(x, t) is sufficiently smooth w.r.t. t for all x ∈ Xand t ∈ T .

Given t ∈ T , denote by Ka(t) ⊂ K the set of active indices in t,Ka(t) := k ∈ K : hTk t = ∆hk, and by L(t) the set of feasible directionsin T starting from t,

L(t) := l ∈ Rs : hTk l ≤ 0, k ∈ Ka(t). (3)

Given x ∈ X , the set of active indices in x is Ta(x) := t ∈ T :f(x, t) = 0.

D e f i n i t i o n 1. Let us say that an index t ∈ T is immobile inproblem (P ), if f(x, t) = 0 for all x ∈ X.

Denote by T ∗ the set of all immobile indices in problem (P ). It isevident that T ∗ ⊂ Ta(x) for all x ∈ X .

D e f i n i t i o n 2. The constraints of problem (P ) satisfy the Slatercondition if there exists x ∈ X such that f(x, t) < 0, ∀t ∈ T.

In [4], it is proved that a convex SIP problem with X 6= ∅ satisfies theSlater condition if and only if the set of immobile indices is empty. Thusthe emptiness of the set T ∗ can be considered as a constraint qualification(CQ) equivalent to the Slater-type condition for SIP.

D e f i n i t i o n 3. An immobile index t ∈ T ∗ has the order ofimmobility q(t, l) along a nontrivial feasible direction l ∈ L(t) if

1. dif(x,t+αl)dαi

∣∣∣α=+0

= 0, ∀x ∈ X, i = 0, . . . , q(t, l),

2. there exists a feasible x ∈ X such that d(q(t,l)+1)f(x,t+αl)

dα(q(t,l)+1)

∣∣∣α=+0

6= 0.

Given t ∈ T, it is easy to see that the set L(t) defined in (3) is aconvex polyhedral cone in R

s. Then, according to the known results onthe convex polyhedral cone’s decomposition (see [3]), there exist a finiteset of vectors bi, i ∈ 1, . . . , p, ai, i ∈ I, such that L(t) admits a finiterepresentation in the parametric form:

L(t) = l ∈ Rs : l =

p∑

i=1

βibi +∑

i∈Iαiai, αi ≥ 0, i ∈ I, (4)

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with p = s−rank(hk, k ∈ K), |I| <∞, βi ∈ R, i ∈ 1, . . . , p, αi ∈ R, i ∈ I.Vectors bi, i ∈ 1, . . . , p satisfy the conditions hTk bi = 0, i = 1, ..., p,

k ∈ K(t), and are usually referred to as bidirectional extremal rays. Vec-tors ai, i ∈ I, in turn, satisfy the inequalities hTk ai ≤ 0, i ∈ I, k ∈ K(t),and are called unidirectional extremal rays. The extremal rays can befound explicitly (see [2]).

R e m a r k 1. In the case of a pointed cone L(t), the set of vectorsbi, i = 1, . . . , p, is empty. If t ∈ int T , then the set of vectors aj , j ∈ I, isempty and bi = ei, i = 1, . . . , p = s.

Suppose now that t ∈ T ∗ ⊂ T is an immobile index in (P ). Considerthe corresponding sets L = L(t), K = Ka(t), and suppose that the ex-tremal rays in L are defined explicitly. Given a sufficiently small ε > 0,denote by Tε(t) an ε-neighborhood of t in T : Tε(t) = t ∈ T : ||t−t|| ≤ ε.From the parametric representation (4) of the cone of feasible directionsL, it follows that the local constraints f(x, t) ≤ 0, ∀t ∈ Tε(t), can bepresented in the form of the following modified constraints:

f(x, (β, α)) ≤ 0, ∀(β, α), α ≥ 0, ||(β, α)|| ≤ ε, (6)

where (β, α)T ∈ Rp+|I|, f(x, (β, α)) := f(x, t+Bβ+Aα), and the columns

of matrices B ∈ Rs×p and A ∈ R

s×|I| are presented by the bidirectionaland unidirectional rays respectively.

Without loss of generality we can use here the maximum norm givenas ||y|| = max

i=1,...,n|yi| for y ∈ R

n. Then the modified constraints (6) can be

considered as the box constraints w.r.t. variables (α, β), α ∈ R|I|, β ∈ R

p.From the definition of the immobility index t, it follows that for any

x ∈ X , the vector t maximizes the function f(x, t), or equivalently, vector(β = 0, α = 0) is a solution of a so called lower level problem:

max(β,α)

f(x, (β, α)), s.t. α ≥ 0. (8)

The first and the second order optimality conditions for the vector(β = 0, α = 0) in problem (8) can be formulated as follows:

∂T f(x, t)

∂tbi = 0, i = 1, . . . , p;

∂T f(x, t)

∂tai ≤ 0, i ∈ I, ∀x ∈ X, (9)

(βT , αT )(B,A)T∂2f(x, t)

∂t2(B,A)

(βα

)≤0, (10)

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for all (βT , αT ) ∈ Rp+|I| such that αi = 0 if ∂f

T (x,t)∂t ai < 0, and αi ≥ 0 if

∂fT (x,t)∂t ai = 0, i ∈ I.A s s u m p t i o n 1. Suppose that X 6= ∅, the set T is bounded and

q(l, t) ≤ 1, ∀l ∈ L(t) \ 0, ∀t ∈ T ∗.It can be showed that Assumption 1 implies the finiteness of the set

of immobile indices: T ∗ = t∗j , j ∈ J∗ with |J∗| < ∞, and the existenceof x ∈ X such that |Ta(x)| <∞.

Suppose that the set of immobile indices and their immobility ordersalong the corresponding extremal rays are known ([5]). Denote:

I∗ := i ∈ I : q(t, ai) = 0 = i ∈ I : ∃x(i) ∈ X :∂T f(x(i), t)

∂tai < 0,

I0 := I\I∗ = i ∈ I :∂T f(x, t)

∂tai = 0, ∀x ∈ X.

Taking into account Assumption 1, we get q(t, bi) = 1, i = 1, . . . , p;q(t, ai) ≥ 1, i ∈ I0; q(t, ai) = 0, i ∈ I∗. Then from conditions (9),(10),

we conclude that for all x ∈ X and (β, α0)T ∈ R

p × R|I0|+ it holds

∂T f(x, t)

∂tbi = 0, i = 1, . . . , p;

∂T f(x, t)

∂tai = 0, i ∈ I0, (11)

∂T f(x, t)

∂tai ≤ 0, i ∈ I∗; (βT , αT0 )(B,A0)

T ∂2f(x, t)

∂t2(B,A0)

(βα0

)≤0,(12)

where A0 = (ai, i ∈ I0), α0 = (αi, i ∈ I0).

Taking into account that for all t ∈ T ∗ and any x ∈ X the relations(11), (12) are satisfied, and repeating the considerations made in [4] forthe case of the box constrained index set T , we prove the following implicitoptimality criterion.

Theorem 1. Under Assumption 1, a vector x0 ∈ X is optimal inthe convex SIP problem (P) with polyhedral index set T , if and only ifthere exists a finite set of indices tj , j ∈ Ja(x

0) ⊂ Ta(x0) \ T ∗ with

|Ja(x0)| ≤ n, such that x0 is optimal in the following auxiliary problem:

(Paux) : minx∈Rn

c(x)

s.t. f(x, tj) ≤ 0, j ∈ Ja(x0),

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f(x, t∗j ) = 0,∂T f(x, t∗j )

∂tB(j) = 0,

∂T f(x, t∗j )

∂tA0(j) = 0,

∂T f(x, t∗j )

∂tA∗(j) ≤ 0,

(βT (j), αT0 (j))(B(j), A0(j))T∂2f(x, t∗j )

∂t2(B(j), A0(j))

(β(j)α0(j)

)≤0,

where (β(j), α0(j))T∈ R

p(j) × R|I0(j)|+ , j ∈ J∗, B(j) = (bi(j), i = 1, ..., p(t∗j )),

and A0(j) = (ai(j), i ∈ I0(t∗j )), A∗(j) = (ai(j), i ∈ I∗(t∗j )).

Notice that the auxiliary problem (Paux) is also a SIP problem but itis more easy to study and solve than the original problem (P ) since

1. the infinite constraints in (Paux) are quadratic w.r.t. multidimen-

sional indices (β(j), α0(j))T∈ R

p(j) × R|I0(j)|+ , hence this problem can be

considered as a light generalization of the common semidefinite (SDP)problem (see [1]);

2. due to Assumption 1, the constraints of (Paux) satisfy the Slatertype condition, i.e. there exists a vector x ∈ X such that for all t∗j ∈T ∗, j ∈ J∗, it is satisfied:

(βT (j), αT0 (j))(B(j), A0(j))T ∂

2f(x,t∗j )

∂t2 (B(j), A0(j))

(β(j)α0(j)

)< 0, (1)

∀(β(j), α0(j))T∈ R

p(j) × R|I0(j)|+ , (β(j), α0(j))

T 6= 0;

3. explicit optimality conditions for SDP-type problems satisfying theSlater condition can be easy formulated and can be efficiently applied totheory and practice of SIP.

The novelty of the approach presented here consists in use of the factthat the immobile indices solve the lower level problem for all feasible x.The analysis of the optimality conditions for the lower level problem allowsone to form a new set of constraints that should be satisfied by all x ∈ X,and to formulate new optimality conditions (implicit or explicit) for theoriginal SIP problem (P) in the form of CQ-free optimality criterion for aspecial auxiliary problem (Paux) that has a more simple structure. Noticethat in the convex case, such new optimality conditions are more strongthan the known ones for SIP (see for example, [1,6]).

The authors were partially supported by the state program ”Mathemat-

ical Models 13” of fundamental research in Republic of Belarus; and by the

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Center for Research and Development in Mathematics and Applications (Uni-

versity of Aveiro) and the Portuguese Foundation for Science and Technology

(“FCT”)(project no. PEst-C/MAT/UI4106/2011).

References

1. Bonnans J.F., Shapiro A. “Perturbation Analysis of Optimization Prob-lems”, Springer-Verlag, New-York (2000).

2. Chernikova N.V. “Algorithm for discovering the set of all the solutions ofa linear programming problem,” U.S.S.R. Computational Mathematics andMathematical Physics, 8, No.6, 282-293 (1968).

3. Fernandez F., Quinton P. “Extension of Chernikova’s algorithm for solv-ing general mixed linear programming problems”, Research Report No 934,IRISA, France, (1988).

4. Kostyukova O.I., Tchemisova T.V. “Implicit Optimality Criterion for ConvexSIP problem with Box Constrained Index Set”, to appear in TOP, (2012).

5. Kostyukova O.I., Tchemisova T.V., and Yermalinskaya S.A. “On the al-gorithm of determination of immobile indices for convex SIP problems”,IJAMAS-International Journal on Mathematics and Statistics, 13, No J08,13-33 (2008).

6. Stein O., Still G. “On optimality conditions for generalized Semi-Infinite Pro-gramming problems”, J.Optim. Theory Appl. 104, N.2,443-458 (2000).

Optimization methods for measurement of returns toscale in the non-radial DEA models

Vladimir Krivonozhko1, Finn Førsund2, Andrey Lychev3

1 National University of Science and Technology “MISiS”, Moscow, Russia;

[email protected] University of Oslo, Oslo, Norway; [email protected]

3 National University of Science and Technology “MISiS”, Moscow, Russia;

[email protected]

The non-radial DEA models [1] possess some specific features. First,multiple reference sets may exist for a production unit. Second, multiplesupporting hyperplanes may occur on optimal units of the frontier. Third,

144

multiple projections (a projection set) may occur in the space of inputand output variables. All these features cause certain difficulties undermeasurement of returns to scale of production units.

The non-radial DEA model can be written in the following form [1, 2]

maxh = (C+TS+ + C−TS−)

subject to

n∑

j=1

Xjλj + S− = Xo,

n∑

j=1

Yjλj − S+ = Yo,

n∑

j=1

λj = 1, S+ ≥ 0, S− ≥ 0, λj ≥ 0, j = 1, . . . , n,

(1)

here Xj = (x1j , . . . , xmj) and Yj = (y1j , . . . , yrj) represent the observedinputs and outputs of production units (Xj , Yj), j = 1, . . . , n, S− =(s−1 , . . . , s

−m) and S+ = (s+1 , . . . , s

+r ) are vectors of slack variables. The

superscript “T” indicates a vector transpose. The components of theobjective-function vectors C+ and C− are specified as follows:

c−k = (m+ r)−1 (maxxkj |j = 1, . . . , n −minxkj |j = 1, . . . , n)−1 ,

c+i = (m+ r)−1 (maxyij |j = 1, . . . , n −minyij|j = 1, . . . , n)−1,

k = 1, . . . ,m, i = 1, . . . , r.

In the model (1), an efficiency score for unit (Xo, Yo) is evaluated, where(Xo, Yo) is any production unit from the set (Xj , Yj), j = 1, . . . , n. Ifthe optimal value h∗ of the model is equal to zero, then unit (Xo, Yo) isconsidered efficient, if h∗ > 0, then the unit is inefficient [1].

Banker et al. [1] proposed a two-stage approach to determine returns toscale in these models. Sueyoshi and Sekitani [2] showed that this approachmay generate incorrect results in some cases. An interesting approach wasproposed for measurement of returns to scale based on using strong com-plementary slackness conditions (SCSC) in the non-radial DEA models

145

(SCSC/NM) [2]. The SCSC/NM model is written in the following form

max

η

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

θXo −n∑

j=1

λjXj ≥ 0,n∑

j=1

λjYj ≥ Yo,n∑

j=1

λj = 1,

λj ≥ 0, j = 1, . . . , n,

vTXo = 1, −vTXj + uTYj + u0 ≤ 0, j = 1, . . . , n,

v ≥ 0, u ≥ 0,

θ = uTYo + u0,

λj + vTXj − uTYj − u0 ≥ η, j = 1, . . . , n,

v −n∑

j=1

λjXj + θXo ≥ η,

u+

n∑

j=1

λjYj − Yo ≥ η, η ≥ 0

. (2)

The first six conditions are from the primal model (1), the next threeconditions are from the dual problem, the tenth condition provides theequality of the objective functions of the primal and dual problems. Thelast three conditions express the SCSC constraints. In order to securethat strong complementarity is obtained the variable η is entered as theobjective function in (2).

Our theoretical consideration and computational experiments showthat the SCSC/NM method may not be efficient from the computationalpoint of view. Model SCSC/NM generates ill-conditioned basic matricesduring the solution process, which results in “strange results” that do notcoincide with the optimal solution of the corresponding non-radial DEAmodel. This naturally contradicts the optimization theory.

In our work we propose a two-stage approach to measure returns toscale in the non-radial DEA models. At first stage, an interior point,belonging to the optimal face, is found using a special elaborated method.In our previous work [3] we proved that any interior point of a face has thesame returns to scale as any other interior point of this face. At the secondstage, we propose to determine the returns to scale at the interior pointfound in the first stage with the help of Banker and Thrall’s method [4]or using the direct method of Førsund et al. [5].

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Our computational experiments documented that the proposed ap-proach is reliable and efficient for solving real-life DEA problems.


(project no. 11-07-00698-a).

References

1. R.D. Banker, W.W. Cooper, L.M. Seiford, R.M. Thrall, J. Zhu. “Returns toscale in different DEA models,” European Journal of Operational Research,154, 345–362 (2004).

2. T. Sueyoshi, K. Sekitani. “Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach,” European Journalof Operational Research, 176, 1918–1946 (2007).

3. V.E. Krivonozhko, F.R. Førsund, A.V. Lychev. “Returns-to-scale propertiesin DEA models: the fundamental role of interior points,” Journal of Produc-tivity Analysis, DOI 10.1007/s11123-011-0253-z (2012).

4. R.D. Banker, R.M. Thrall. “Estimation of returns to scale using Data En-velopment Analysis,” European Journal of Operational Research, 62, 74–84(1992).

5. F.R. Førsund, L. Hjalmarsson, V.E. Krivonozhko, O.B. Utkin. “Calculationof scale elasticities in DEA models: direct and indirect approaches,” Journalof Productivity Analysis, 28, 45–56 (2007).

Selection of the target audience by the leveragemethod in the expert system for advertising specialist

Ulyana Kulbida1, Olga Kaneva 2

1 Omsk State Technical University, Omsk, Russian; uni [email protected] Omsk State Technical University, Omsk, Russian [email protected]

Advertising positioning is an independent branch within the generaltheory of positioning, and it determines the optimal way of presentinginformation about a trademark to a particular target audience.

The market position of a trademark defines the location of goods onthe market, while the advertising position defines the individuality of the

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trademark in the communications area. Well-known researchers of mar-keting and advertising Rossiter and Percy proposed to conduct advertisingpositioning of the trademark in levels: macro-level (X − Y Z), meso-level(I −D− U), micro-level (a− b− e) [1]. Further we are going to examinethe above mentioned levels more detailed.

Macro-level. This level determines the place of the trademark withinappropriate product category, selects a target audience and the way ofthe trademarks positioning, taking into account particular qualities of theconsumer or the product. The scheme of macro level can be expressed bythe following formula: “Product X provides to people Y assistance Z”.In this formula X equals the trademark, Y equals the target audience,while Z equals the benefits of the trademark.

Meso-level. On this level, it is to be decided what benefits should bedetermined while positioning (the model I−D−U). Benefits emphasizedin advertising, should correspond with the following three main conditions:

1) Importance: correspondence between the benefits and the motivethat drives the one who buyes the trademark.

2) Delivery: the consumer’s subjective opinion about the trademarksability to provide benefits.

3) Uniqueness: the perceived ability of the trademark to provide ben-efits better than other trademarks do.

Micro-level. This level defines the way of focusing on the main bene-fits. The technique of focusing on the benefits of the trademark is basedon distinguishing characteristics (physical properties) of the product, itsbenefits (what the buyer wants) and emotions (feelings of the buyer causedby buying or by using the product). The model was named after the firstletters of these terms a− b − e, in which a stands for attribute, b standsfor benefit, and e stands for emotion (e+ positive, e− negative).

Making decision on the macro-, meso- and micro-levels of positioningallows us to make a conclusion about the position of the trademark. Thisconlusion presents the trademarks creative strategy. Obtaining a detailedscheme of such a conclusion is the decision of the unformalized problemin the field of advertising.

Work “Expert system for positioning and brand advertising strategies”[2] proposes the algorithm of the solution of the first problem of the macro-level (determination of the trademarks place within appropriate productcategory) for the developed expert system. This algorithm is based onthe use of attractiveness and competitiveness indicators. The results of

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the numerical experiments are given.The next problem of the macro-level is the selection of the target

audience. Target audience is a group of people to which the specificcampaign advertising or sales promotion is aimed at [1]. According to theRossiter and Percy there are five groups of costumers:

1) New consumers of the product category. This group of people buy-ing our product, gets acquainted with the category.

2) Loyal consumers, who buy our product on a regular basis.3) Fickle consumers, who buy our products as well as other brands.4) Fickle consumers of other brands, who buy products from other

brands, but not ours.5) Loyal to another brands consumers, who regularly buy someone

else’s product brand.Generally the target audience it is only one group of consumers. How-

ever, sometimes it is useful to define the primary and secondary targetaudiences.

Frequently, the purpose of the advertising campaign is not only toattract new customers from other groups, but to keep people who arealready loyal to the brand. The group of loyal customers of the ownbrand it is usually the secondary target audience of new campaigns whichare aimed at other groups of buyers.

It is necessary to estimate distributing capacity (the quantity of itemsor services that the company can sell) of each company using the followingcorrelation: potential growth in sales (in monetary terms) to the value ofthe event of the advertising campaign, which can provide this growth insales. This correlation is called “leverage”. If we know the number of thecustomers in the certain group and the cost value for realization of theadvertising campaign in this group we can count the common value of theleverage for the group:

Profit leverage =number of the customers× growth of profit

cost value of the advertising campaign. (1)

Obviously, in the usual case a group of customers can be consideredas the target audience only if the profit leverage of this group exceedsone unit. This means that the effect of the advertising campaign willsurpass the cost of its implementation. The higher profit leverage, thelarger group of buyers corresponds to the role of the target audience. Thecharacter of the changing of the leverage with time is almost equal for the

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third and fourth groups of customers, consequently leverage is calculatednot for five, but four groups of buyers. On the assumption of the valuesof the leverage we can determine the primary and sometimes secondarytarget audiences.

Cost calculation of the advertising campaign (the values in the denom-inator of the formula (1)) is a nontrivial problem. Lots of professionals arebased on personal experience, common sense and simple interdependencesin the calculation of the advertising budget. Recently there were more so-phisticated methods of calculating the advertising budget, but also theirevaluation and application should not be divorced from practice.

In the developed expert system, there is an approach that uses theindex of MEF - the minimum effective frequency (the model of the op-timization of the minimum effective frequency of Rossiter-Danaher). Theindex MEF shows the number of the consumers contacts with the ad-vertisement, which is optimal for the influence on the target audience.Measured in absolute units [1]:

MEF = 1 +AMA× (TA+BA+ CA+ PI), (2)

where 1 – initial level MEF in one advertising contact;AMA – corrective factor “attention to the means of advertising”;TA – corrective factor “target audience”;BA – corrective factor “brand awareness”;CA – corrective factor “character of the advertisement”;PI – corrective factor “personal influence”.Table 1 gives the approximate values for the quantities involved in the

calculating of MEF .

The algorithm for calculating of the cost of the campaign1) Let i – number of the customers group (i = 1, 4), j – number of

advertising media vehicle which is used (j = 1,m).2) Define the vectors: α = (α1, . . . , αm)T and β = (β1, . . . , βm)T ,

where αj – cost of the developing promotional material for the mediavehicle j, βj – the cost of placement of the one advertising exposure atthe media vehicle j.

3) For each group of customers are counting the value of νi =MEFi,i = 1, 4, whereMEFi – the minimum effective frequency for i-th group ofcustomers (2). In the result we get vector quantity ν = (ν1, ν2, ν3, ν4)

T .

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4) Let xij – number of the advertising exposure at j-th media vehiclefor i-th target audience. We obtain a matrix of variables:

X =

x11 x12 . . . x1mx21 x22 . . . x2mx31 x32 . . . x3mx41 x42 . . . x4m

.

5) To determine the advertising budget of each customer group it isrequired to solve the following problem of integer programming:

m∑

j=1

βjxij → min, (3)

m∑

j=1

xij ≥ νj , xij ≥ γj , xij ∈ Z, j = 1,m,

where γj ≥ 0 – the prescribed values, indicating the required number ofadvertising exposures at the j-th media vehicle for the i-th target au-dience. The quantity on the objective function is equal to the cost ofplacement of the advertising at the media vehicles for the i-th group ofcustomers.

6) After solving integer programming problems for each group of cus-tomers we will get the values of C1, C2, C3 and C4 campaign by the for-mula:

Ci =

m∑

j=1

βjxij +

m∑

j=1

αj .

Then we use the formula (1) for the counting the value of leverage foreach customer group and determination which of the groups will be thetarget audience.

In the future we plan refinement algorithm by entering into the model(3) indicators of the effectiveness of advertising on each media vehicle.

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Table 1: The values of correction factors

Value of the factorFactor 0 1 2 3

Attentionto themeans ofadvertising

Strong atten-tion

Lowattention

Target au-dience

Loyalcon-sumersof ourbrand

Fickle con-sumers ofour brand

Consumersof otherbrands

New users of theprodukt

Brandawareness

Advertisementfor brandrecognition

Advertisementfor brand recall

Characterof the ad-vertisement

Informationadvertise-ment

Transformationaladvertisement

Personalinfluence

Strongper-sonalinflu-ence

Low personalinfluence


(project no. 12-07-00326).

References

1. J.R. Rossiter, L. Percy. Advertising and promotion of products, Piter Pub-lishing House, St. Peter (2001).

2. A.V Zykina, O.N. Kaneva, U.N. Kulbida. “Expert system for positioningbrand advertising strategies”, Omsk Scientific Bulletin, No. 3(103), 253–257(2011).

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Zonal control of lumped systems on different classesof feedback functions

Samir Kuliev1

1 Cybernetics Institute of ANAS, Baku, Azerbaijan; [email protected]

In this work we investigate a class of feedback control problems for dy-namic, in the general case, nonlinear objects involving lumped parameters.For synthesized control actions we introduce the notion of zonality thatmeans constancy of the synthesized control parameters’ values in each ofthe subsets (zones). These subsets are obtained by partitioning the set ofall possible states of the object investigated. Formulas for the gradient ofthe target functional with respect to the optimizable parameters of thesynthesized controls are obtained. These formulas can be used to buildnumerical solution schemes on basis of first order iterative optimizationmethods. Results of numerical experiments obtained by solving some testproblems are given.

Let the controlled process be described by the following nonlinear dif-ferential equations system:

x(t) = f(x(t), u(t), p), t > 0, (1)

x(0) = x0 ∈ X0 ⊂ Rn, p ∈ P ⊂ Rm, (2)

where x(t) is the n-dimensional vector function of the process state; u(t) ∈U is the r-dimensional control vector function; U ⊂ Rr is the closed setof the control actions’ admissible values; p is the m-dimensional vectorof the process’s constant parameters, the values of which are uncertain,but there is a set of their possible values P and the density (weighting)function ρP (p) ≥ 0 defined on P ; X0 is the set of possible values of theprocess’s initial states with the density (weighting) function ρX0(x0) ≥ 0given.

Control of the process (1) is realized with the use of feedback; thestate vector x(t) may be measured fully or partially. Observations of theprocess state may be carried out at discrete points of time or continuously.To control the process, we propose to choose the values of the synthesizedcontrol actions according to a subset (zone) the measured current processstate belongs to. The subsets are obtained by partitioning the set of allpossible phase states of the object.

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The objective of the feedback control problem considered is to de-termine the values of the parameters of the zonal control actions u(t)minimizing the following functional

J(u) =

∫

X0

∫

P

I(u, T ; x0, p)ρX0(x0)ρP (p)dPdX0/(mesX0 ·mesP ), (3)

I(u, T ; x0, p) =

T∫

0

g(x(t), u(t))dt+Φ(x(T ), T ). (4)

Here x(t) = x(t; x0, p, u) is the solution to the system (1) under theadmissible control u(t), initial state x0, and the values of the parametersp; T = T (x0, P ) is the corresponding completion time of the process,which can be either a fixed quantity T = T (x0, P ) = const = T , or anoptimizable function of the values of the initial state and of the object’sparameters T =

T (x0, P ) : T (x0, P ) ≤ T , x0 ∈ X0, p ∈ P

, where T is

given. The latter case arises in, as a rule, speed-in-action problems forcontrol systems. We considered both cases.

The functional (3) and (4) defines the quality of control which is op-timal on the average with respect to the admissible values x0 ∈ X0 andp ∈ P . Denote by X ⊂ Rn the set of all possible states of the object underdifferent admissible initial states x0 ∈ X0, the values of the parametersp ∈ P , and the controls u(t) ∈ U for t ∈ [0, T ].

Let the set X be partitioned into given number L of open subsets X i

such that

L

Ui=1

Xi= X, Xj ∩X i = ∅, i 6= j, i, j = 1, 2, ..., L,

where Xiis the closure of the set X i. In the work we consider the follow-

ing four types of feedback control problems, which differ in organizationof feedback with the object and, therefore, in formation of the controlactions’ values.

P r o b l e m 1. There are points of time τj ∈ [0, T ], j = 0, 1, ..., N ,τ0 = 0 given, at which it is possible to observe the current state of theprocess x(τj) ∈ X . The frequency of these observations is such that whenthe process state belongs to some subset, it is observed at least once.The values of the control u(t), which are constant for t ∈ [τj , τj+1), are

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assigned depending on the value of the last observed current process state,namely, depending on the subset X i, i = 1, 2, ..., L of the phase space Xwhich the measured (observed) current state belongs to. Therefore

u(t) = vi = const, x(τj) ∈ X i, t ∈ [τj , τj+1), (5)

vi ∈ U ⊂ Rr, i = 1, 2, ...L, j = 0, 1, ...N − 1, τN = T .

It is required to determine zonal values of the control vi, i = 1, 2, ...L,optimizing the functional (3).

P r o b l e m 2. The control actions are defined in the form of alinear function of the results of observations of the state variables at givendiscrete points of time τi ∈ [0, T ], i = 0, 1, ..., N :

u(t) = Ki1 · x(τj) +Ki

2, t ∈ [τj , τj+1), x(τj) ∈ X i, t ∈ [0, T ],

i = 1, 2, ..., L, j = 0, 1, ..., N − 1. (6)

HereKi1 is the (r × n) matrix andKi

2 is the r-dimensional vector whichare constant for t ∈ [τj−1, τj). The problem is to determine the valuesKi

1, Ki2, i = 1, 2, ..., L, optimizing the functional (3).

P r o b l e m 3. Continuous observation of the process state is carriedout; the control actions take zonal values of the control:

u(t) = wi = const, x(t) ∈ X i, t ∈ [0, T ],

wi ∈ U ⊂ Rr, i = 1, 2, ..., L. (7)

It is required to determine the zonal values of the control wi, i =1, 2, ...L, optimizing the functional (3).

P r o b l e m 4. Continuous observation of the process state is carriedout; the control actions are defined by a linear function of the measuredcurrent values of the process variables:

u(t) = Li1 · x(t) + Li2,

x(t) ∈ X i, t ∈ [0, T ], i = 1, 2, ..., L, j = 0, 1, ..., N − 1. (8)

Here Li1 is the (r × n) matrix and Li2 is the r-dimensional vector whichare constant for each subset X i, i.e. while x(t) ∈ X i. It is required todetermine the values Li1, Li2, i = 1, 2, ..., L, optimizing the functional (3).

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Note that in all the four problems, the synthesized controls are definedby the finite-dimensional constant vectors and matrices.

To solve the optimization problems stated above numerically and todetermine the control actions from the classes (5)-(8), we propose to usefirst order optimization methods and the corresponding standard software[1]. For this purpose, we obtained formulas for the gradient of the targetfunctional using the technique of the target functional increment obtainedat the expense of the optimizable arguments increment [2]. Results of nu-merical experiments carried out by the example of the solution to severalmodel problems are given.

References

1. N.Jorge, S.J. Wright. Numerical Optimization, Springer-Verlag, New-York,Inc. (1999).

2. F.P. Vasilyev. Optimization methods, Factorial Press, Moscow, (2002).

3. S. Z. Kuliev. “Synthesis of zonal controls of nonlinear systems under discreteobservations”, Automatic Control and Computer Sciences, Allerton Press,45, No. 6, 338–345 (2011).

The Heuristic Approach to movement optimizationon single-track part of the railway net

Maya Laskova1, Alexander Lazarev2, Elena Musatova3

1 The Institute of Control Sciences V. A. Trapeznikov Academy of Sciences,

Moscow, Russia; [email protected] The Institute of Control Sciences V. A. Trapeznikov Academy of Sciences,

Moscow, Russia; [email protected] The Institute of Control Sciences V. A. Trapeznikov Academy of Sciences,

Moscow, Russia; [email protected]

Abstract: This paper represents our solution for the problem of move-ment organization based on timetable optimization on the problematicpart of railway system, i.e. single-track line. The approximate solutionof this problem was founded on the heuristic method. The method givesthe exact results in the case of limited amount of parameters and also can

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be used in the case with huge number of parameters due to reasonablecomputational time.

Key words: scheduling theory, algorithm, single-track railway problem.

Introduction

Single track railways are of great interest to scheduling theory becausenowadays they are the most weak chain in the railroad transportationall over the world and especially in Russia. The article tend to solvingbottleneck problems in transport network. Bottleneck is a part of the waywith low bandwidth compare with other parts of the same road. Oftenit is the railway line with a limited number of tracks. It can also be anarrow bridge, tunnel or narrow causeway. Presence of such railway partsoften cause delays and Timetable failure, and it is necessary to obtaingood solution by optimization of schedule (Timetable). This task withsingle line track is known to be NP-hard; we made an attempt to create anHeuristic algorithm which will help to minimize reasonable computationaltime for solving this problem.

Problem formulation

We consider that arrival numerouse applications to stations 1 and 2are known in advance. Considering the trains with following parameters:N = N1 ∪N2 — set of trains;N1 = 1, 2, ..., n — set of trains arrived at the station 1;N1 = 1, 2, ...,m — set of trains arrived at the station 2;r1i — planning time of i train to station 1;r2j — planning time of j train to station 2;

d1i — the due date of arrival i train i ∈ (1, n) at the station 1 to thestation 2;d2j — the due date of arrival j train j ∈ (1,m) at the station 2 to thestation 1;p — the average time of movement the train (p = const);δ — headway between trains.

Fact data

C1i — real arrival time of i train to station 2;

C2j — real arrival time of j train to station 1;

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S1i — real departure time of the i train;S2j — real departure time of the j train.

The objective function is following: T∑(π) =N∑i=1

max0, Ci − di

Our task was to create the train schedule optimizing the total delayof movement trains on the single-track part of the railway net.(Fig.1)

Fig. 1. Bottleneck problem

Note 1. If δ = p ,i.e. on the railway at the time can be no morethan one of train. The problem reduces to (m + n) service requirementson a single device to the agreed arrival time and the due date. There arenumber of polynomial algorithms for solving such problems [1-3].

Note 2. Further, we assume that δ = 0 , i.e. the delay time of eachtrain is negligible compared to the distance between stations.

The heuristic algorithm

The algorithm consists of two parts: direct flow and indirect flow.

D e f i n i t i o n 1. Batch(n, i) — a set, containing n trains, departedfrom the station 2 at the same moment Cin = maxr2n, r1i + p.

Under the direct flow we create Batch(n, i) (see def.1). For Batch(n, i)we determine all possible departure moments Sin. In this way the trainsfrom the station 1 with r1i ∈ (Sin − p;Sin + p) will depart at the momentt = Sin + p. At this stage we anylize the getting results of objectivefunction F (Skn); and afterwords we choose the smallest meanning. Thus

158

we got the optimal departure time Sn for the least train. Similary we findthe optimal departure time Sn−1, Sn−2, ...S1.

D e f i n i t i o n 2. Batch(m, j) — a set, containing m trains,departed from the station 1 at the same moment Sjm = maxr1m, r2j + p,j ∈ (1, n).

In the calculation of the indirect flow we create Batch(m, j).The development of this algorithm alows to simplify the task of sched-

ule optimization and it is especially use for failure movement.

References

1. A.A. Lazarev, E.R. Gafarov. The scheduling theory. Minimization of totaltardiness for one machine, M.:RAC, Moscow (2006).

2. A.A. Lazarev, E.R. Gafarov. The scheduling theory. Problems and algo-rithms, M.:MSU, Moscow (2011).

3. E.R. Gafarov, A.A. Lazarev, F. Werner. Transforming a pseudo-polynomialalgorithm for the single machine total tardiness maximization problem intopolynomial one, DOI 10.1007/slo479-011-1055-4.

Finite time-interval robustness study of dynamicsystems with imprecisely identified parameters

Alexander V. Lotov1, Georgij Kamenev2

1 Dorodnicyn Computing Center of RAS, Moscow, Russia; [email protected] Dorodnicyn Computing Center of RAS, Moscow, Russia; [email protected]

New technique for analysis of dynamic models of complex systems(economic, biological, etc.) is proposed. The models are characterizedby ambiguity of behavior because of inaccuracy of identification of theirparameters. In this case, quantitative analysis of the robustness of trajec-tories of the model at finite time intervals with respect to imprecision ofparameters is needed. It is based herein on approximating the trajectorytubes. The robustness study can be used, for example, in the frameworkof multi-criteria decision support. The technique is based on methodsfor approximating the reachable sets of dynamical models and methods

159

for approximating the feasible criterion set in multi-criteria problems (see[1]) as well as on application of these methods in visual identification ofparameters [2].

Let the dynamics of the system under study to be described by thesystem of difference equations

(xk+1 − xk)/τ = f(xk, uk, ck), k = 0, . . . , N − 1, (1)

where xk ∈ Rn is the state vector, uk ∈ Rr is the control vector, ck ∈ Rsis the vector of unknown parameters, at the time-moment k. The initialstate x0 = x0 is assumed to be given. The technique consists of two steps.

Step 1. Parameter identification. It is assumed that there is informa-tion about the behavior of the dynamical system under study in the pastthat allows to draw conclusions about possible values of the parametervector c. The parameter identification method is based on visualizationof the multi-dimensional graph of error function [2], which helps to ana-lyze the stability of the solution of the error function minimizing problem.If the solution is not stable, the expert points out such a region C in theparameter space Rs, that the solution of the parameter identification hasthe form c ∈ C. In such an approach, the model parameters are identi-fied by using a synthesis of observations and non-formal experience of theexpert. Further, the study examines the case when the region C containsmore than one point.

Step 2. Stability analysis2a. Let us consider the case when the vector c ∈ C is constant in

time, i.e. ck = c, k = 0, . . . , N − 1. In this case, for a given controluk, k = 0, . . . , N − 1, the system (1) allows constructing the trajectory ofthe system for each vector c ∈ C. The output for the entire set C, i.e.the set of uncertainty, can be approximated for the time moment k = Nas well as for any intermediate time moment. It can be done using themethod for approximating the set of feasible objective values for nonlinearsystems described in [1]. The method is based on the constructing thetrajectories for a large number of random vectors from C and furthercovering the set of the ends of the trajectories, i.e. the points xN , by arelatively small number of parallelotops, i.e. polytopes which hyperplanesare parallel to the coordinate planes. The method is provided with anestimate of the quality of such an approximation.

The constructed sets of uncertainty are approximations of cross-sectionsof the trajectory tubes of the system (1) while c ∈ C with the given control

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uk, k = 0, . . . , N − 1. In total, they approximate the trajectory tube ofthe system (1), which we denote by X . Visualization of the cross-sectionsallows quantitative analyzing the stability of the trajectories of the sys-tem (1) with respect to the uncertainty in the parameter vector c ∈ C. Itsupports evaluating the possibility of using the tube of trajectories in thedecision problem of the selecting the control.

2b. Let the vector c change in time. Let us consider a possible de-viation of a trajectory from the ”unperturbed” vector-valued functionck, k = 0, . . . , N − 1 under the condition c ∈ C. If it is not too large, it ispossible to use the linearization of the equation (1). Suppose, moreover,that the set C is convex. For given function uk, k = 0, . . . , N − 1 andck, k = 0, . . . , N − 1, the trajectory of (1) is constructed, which we denoteby xk, k = 0, . . . , N . Then, the equation (1) is linearized in the neighbor-hood of xk, k = 0, . . . , N , uk, k = 0, . . . , N − 1 and ck, k = 0, . . . , N − 1.Next, by using the methods of polyhedral approximation of the reachablesets in the convex case (see [1]), a collection of sets of uncertainty for thelinearized system for N steps, as well as for intermediate points in time, isconstructed. In the same way as in the case 2a, this set of approximationsallows us to study the tube of trajectories of the linearized system, whichis also denoted by X . Just as in the previous case, the visualization ofthe cross-sections allows to quantitatively analyze the stability of the tra-jectories of the system with respect to the uncertainty in the parametervector c and to evaluate the possibility of using the tube of trajectorytubes in the decision problems.

Assume that the difference system turned out to be sufficiently ro-bust to the imprecision of parameters. Let us consider a decision problemcharacterized by m criteria, denoted by y and a finite number controlalternatives u1, . . . , uM where uj = (ukj , k = 0, . . . , N − 1). For the sim-

plicity, let us assume that y = f(xN ) and c ∈ C does not change in time.By using the method for approximating the feasible set of criterion vectorsfor nonlinear systems mentioned in section 2a, one can approximate thefeasible criterion set Y (j) related to any given control uj and all possiblevectors c ∈ C. By his the problem is reduced to selecting a control on thebasis of comparing the sets Y (j), j = 1, . . . ,M . Selecting a control can bebased on the reasonable goals method [1]. For the case of the impreciseoutputs of alternatives, this method was proposed in [3] and studied in [4].In the framework of the method, the convex hull of the best points of the

161

polyhedra that approximate the sets Y (j), j = 1, . . . ,M , is constructed.The interactive visualization of the convex hull is used for studying itsmulti-dimensional Pareto frontier. By specifying the most preferred pointof the Pareto frontier (reasonable goal), the user expresses his/her prefer-ences. Then, a small number of such alternatives is found that are in linewith the specified goals.


(projects no.10-01-00199 and 11-01-12136-ophi-m) and Programs of fundamen-

tal research of Russian Academy of Sciences Pi-15 and Pi-18.

References

1. A.V. Lotov, V.A. Bushenkov, G.K.Kamenev. Interactive Decision Maps. Ap-proximation and Visualization of Pareto Frontier, Kluwer, Boston (2004).

2. G.K.Kamenev. “Visual identification of parameters of models in the case ofsolution ambiguity,” Mathematical modeling, 22, No. 9, 116–128 (2010).

3. A.V. Lotov. “Visualization-based Selection-aimed Data Mining with FuzzyData,” International Journal of Information Technology and Decision Making,5, No. 4, 611–621 (2006).

4. A.V. Lotov, A.V. Kholmov. “Reasonable goals method in the multi-criteriachoice problem with uncertain information,” Doklady Mathematics, 80, No.3, 918–920 (2009).

Estimation of economic damage from humanmortality by external causes on macro-, meso-, micro-

levels

A.A. Lukovenko1, T.M. Tikhomirova2

1 Plekhanov Russian University of Economics, Moscow, Russia;

[email protected] Plekhanov Russian University of Economics, Moscow, Russia;

t [email protected]

In modern Russia there is no universal accepted methods for estima-tion of economic damage from human untimely mortality. Damage from

162

mortality by external causes may be evaluated in particular as: GDP peremployed person, annual salary, consumption, etc. But in practice usingany methods leads to a number of difficulties and criticisms, which essen-tially is that every economic level has its own goals and objectives thataffect the incoming flow of socioeconomic factors.

In order to estimate economic damage from untimely mortality it isnecessary to analyze contributing factors and develop particular methodsinherent in every level, which may work as in the system as separately forindividual appraisals.

It is worth to mentioning that on macrolevel the estimation of damagefrom mortality by external causes should be maximum aggregated andcorrelated with social prosperity of the country or group of countries. Theestimation will allow to determine the amount of short-received GDP dueto disposal of the economically active population and to reflect the amountof social benefits and payments in the event of unexpected death.

On meso-level definition of economic damage estimation affords toreflect the system of measures in order to level differentiation betweenregions on the socioeconomic development, also improve the targeting toallocate investments, required to improve the living standards of regions.

On microlevel the damage from unexpected human death should bedetailed to the level of an objective appraisal of individual life insurance.

Depending on the level the estimation of economic damage may becharacterized by the influence of a various set of factors, such as: hu-man life cost, death rate by external causes, inflation, crime, activitiesrisk-bearing for life - on macrolevel; gender and age population struc-ture, marriage and divorce rates, migration, educational level, HumanDevelopment Index - on meso-level; gender, age, marital condition, areaof employment, chronic diseases, income level, stress level in region ofresidence, educational level - on microlevel.

The application of economic and mathematical approaches and mod-els plays an important role in development of methods for definition ofeconomic damage from external causes for every level. For macrolevelapptication of factor models is most characteristic, where the estimation

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of damage is influenced by macroeconomic and socio-demographic factors.

On meso-level object of observation is a city, region or group of regions.The estimation of damage strongly depends on both the microeconomicfactors that reflect the geographical features and the specifics of the re-gion and the macroeconomic factors that regulate the economy in theregion. To develop methods for economic damage definition there will berequired a probabilistic model, where the independent variables will besocioeconomic factors.

On microlevel object of observation is a person or group of people,so that all the influencing factors are taken into account with a certainprobability, which allows the application of probabilistic models.

References

1. D. Bogoyavlenskiy “Lost years of potential life,”[electronic recource]: Demo-scope Weekly, 1, No. 29-30, (2001).

2. V.I. Orlov “Differentiated Estimation of health and economic losses due tountimely mortality”, Diss...C.m.s., FRI HOI, Moscow, (2009).

3. D.E. Shmakov “Estimation of economic damage from population health lossesin Russia and its regions”, Diss...C.e.s., IEF RAS, Moscow, (2004).

Statistical Methods and Optimization in Data Mining

Eloısa Macedo1, Adelaide Freitas2

1 University of Aveiro, Aveiro, Portugal; [email protected] University of Aveiro, Aveiro, Portugal; [email protected]

The increasing number of the sequenced genomes has created newchallenges in several scientific domains, namely statistics, optimizationand computer sciences. Various numerical transformations related to thesequenced genomes (e.g., frequency of each nucleotide, association be-tween consecutive genomic symbols) have been proposed in order to takeadvantage of statistical methodologies available for quantitative data. It is

164

expected that such numerical data sets contain useful information aboutmathematical properties of DNA sequences. An important issue asso-ciated with data sets where each individual is characterized by a high-dimensional vector of variables consists in the identification of patterns orhomogeneous groups. Since high dimensionality turns the visualizationand analysis of data into a complex problem, the space reduction andthe features subset selection techniques are aimed to facilitate the visual-ization and capture the important and relevant relationships existing indata.

To detect the existence of patterns in a data matrix (n objects × pvariables), it is often desirable to partition the data sets according tosome similarity criteria. This task is related to the data mining tech-nique of partitioning data sets into groups of objects with some similarproperties (clusters) called clustering. There exists a variety of clusteringtechniques designed for several data types, applied in many areas suchas pattern recognition, image segmentation and bioinformatics [1,4,5].Clustering problems are usually formulated as mixed-integer problems, or(0, 1)-semidefinite and semi-infinite programming problems that in turncan be reduced to nonsmooth and nonconvex nonlinear problems [2,4].

While dimensionality reduction of objects is usually achieved by clus-tering techniques, the dimensionality reduction of the variable space canbe provided applying statistical techniques such as Principal ComponentAnalysis (PCA), to detection of a lower number of uncorrelated vari-ables (components) able to explain the maximum variability of the data.The reduction of objects and variables can be obtained applying the twotechniques sequentially. Recently, a new technique called Clustering andDisjoint Principal Component Analysis (CDPCA) was suggested in [3] tosolve the clustering of objects and the partition of variables using PCAsimultaneously. This technique permits to cluster objects along a set ofcentroids and to partition of variables along a reduced set of components,in order to maximize the between cluster deviance of the components inthe reduced space. The model obtained is a quadratic mixed continuousand integer optimization problem. In [3], this model is solved by an alter-nating least-squares (ALS) algorithm that can be considered as an heuris-tic that divides the model solving iteratively in four steps, modifying ineach step certain parameters of the data. The methods of Mixed-IntegerProgramming are used on the basic steps of the algorithm. In [3], theCDPCA algorithm was tested for two data sets, one with 20 objects and

165

6 variables and other with 103 objects and 12 variables.The main objective of this work is to test the ability of this new tech-

nique on biological data sets to make possible visual representation ofrelevant characteristics for data interpretation. For this purpose, we im-plemented CDPCA in R language, which is an open source software widelyused in statistics, with a lot of specific packages for efficient data treatment[6].

Let us introduce the notations that will be used.— X = [xij ] is the data matrix with I objects and J variables (variablesare supposed to be normalized);— P , Q are the desirable numbers of clusters of objects and variables,respectively;— E is the I × J error matrix;— U = [uip] is a I ×P binary matrix and row stochastic defining a parti-tion of objects into P clusters where uip = 1 if the i-th object belongs tocluster p, otherwise, uip = 0;— V = [vjq ] is a J ×Q binary matrix and row stochastic defining a parti-tion of variables into Q clusters where vjq = 1 if the j-th variable belongsto cluster q, otherwise, vjq = 0;— A is the J×Q matrix of the coefficients of the linear combination, suchthat rank(A) = Q and each row (variable) contributes to a single column(component);

— Y = [yiq =∑J

j=1 ajqxij ] is the I × Q component score matrix whereyjq is the value of the i-th object for the q-th component yq (commoninformation of a subset of variables);— X is the P × J matrix of individual centroids in the space of the ob-served variables;— Y is the P ×Q matrix of individual centroids in the reduced space.

The model associated to CDPCA minimizes the norm of the errormatrix

E = X − UY ATw.r.t. parameters representing U , Y and A subject to certain constraints.According to [3], A can be decomposed in the form A = BV , where B isa J × J diagonal matrix of the form

B =

Q∑

q=1

diag(vq)diag(cq),

166

where vq is the vector corresponding to column q in matrix V and cq isthe eigenvector associated to the largest eigenvalue of the matrix

diag(vq)XTUTUXdiag(vq).

We can formulate the problem as follows.

maxU,X,B,V

‖UXBV ‖2 = maxv,c,x,u

P∑

p=1

Q∑

q=1

J∑

j=1

vjqcqxpj

2I∑

i=1

uip

s. t.

P∑

p=1

uip = 1, uip ∈ 0, 1 , i = 1, ..., I; p = 1, ..., P,

Q∑

q=1

vjq = 1, vjq ∈ 0, 1 , j = 1, ..., J ; q = 1, ..., Q, (P )

J∑

j=1

c2jq = 1, q = 1, ..., Q,

J∑

j=1

cjqcjr = 0, q = 1, ..., Q− 1; r = q + 1, ..., Q.

The alternating least-squares algorithm suggested in [3] alternates fourbasic steps: update V (allocation of variables), update B (the PCA step),update U (allocation of objects) and update X (centroid matrix), and itis summarized in the following box. Here, the estimates of the matricesare denoted by .

167

ALS Algorithm for CDPCAinput: numeric data matrix X and tolerance ε

Generate (e.g. randomly) U and V , considering the constraints ofproblem (P);

Compute ˆX =(

UT U)

−1

UTX. Set k=1;

while Fk+1(B, U , ˆX, V )− Fk(B, U , ˆX, V ) < ε:

Update B: Given ˆX, U , V , calculate B =Q∑

q=1

diag(vq)diag(cq).

Update V : Given B, ˆX, U , for j = 1, ..., J , set:

vjq =

1, if F (cq , U , ˆX, [vjq ]) = maxr=1,...,Q

F (cr, U , ˆX, [vjr = 1])

0, otherwise.

where F (cq, U , ˆX, V ) = ‖U ˆXBV ‖2.

Update U : Given B, ˆX, V , for i = 1, ...I , set:

uip =

1, if ‖V T Bxi − V T B ˆxp‖2 = min

s=1,...,P

‖V T Bxi − V T B ˆxs‖2

0, otherwise.

Update X: Given B, U , V , calculate ˆX =(

UT U)

−1

UTX.

Compute Fk(B, U , ˆX, V ) = ‖U ˆXBV ‖2.

do k = k + 1;

The algorithm stops when the difference between consecutive compu-tations of the values of the objective function of problem (P) is smallerthan a specified threshold ε > 0. According to [3], since F (B,U, X, V ) isbounded above, the algorithm converges to a stationary point, which isa local maximum of problem (P). To guarantee that the algorithm findsthe global minimum, the authors of the heuristic in [3] suggest to applythe algorithm repetitively for different initial values of matrices U and V ,that are randomly chosen.

In order to test the ability of the CDPCA to reveal and visualizebiologically meaningful patterns in a 2-dimensional reduced space, we haveimplemented the algorithm using R and carried out an experimental studyinvolving several real data sets extracted from molecular biology domain.Besides the matrices U , V , A, the implementation of CDPCA suggested inthis work returns a pseudo-confusion matrix and draws two scatterplotswhere the data are displayed in the 2-dimensional reduced space, onewhere the objects are labelled according to the real classification and other

168

with the classification found by CDPCA. The pseudo-confusion matrixindicates the number of objects introduced in each cluster (the real andthat found by CDPCA).

On the basis of the realized numerical tests we conclude that the im-plementation of the CDPCA algorithm in R is efficient for the tested datasets. The main advantage of this technique is that each component ischaracterized by a disjoint set of variables. This offers a promising ap-proach for the clustered visual representation of data. On the other hand,it permits to overcome the difficulties on the interpretability of the data inthe reduced space. The proposed heuristic can be improved, since we canupdate the parameters of problem (P) simultaneously using optimizationmethods that efficiently use the structure and properties of this problem.This is a subject of further research.

This work was supported by FEDER founds through COMPETE–Opera-

tional Programme Factors of Competitiveness and by Portuguese founds through

the Center for Research and Development in Mathematics and Applications

(CIDMA, University of Aveiro) and the Portuguese Foundation for Science and

Technology (FCT).

References

1. A. Freitas, V. Afreixo, M. Pinheiro, J. L. Oliveira, G. Moura, M. San-tos. “Improving the performance of the iterative signature algorithm for theidentification of relevant patterns,” Statistical Analysis and Data Mining, 4,No. 1, 71–83 (2011).

2. J. Peny, Y. Wei. “Approximating K-means-type clustering via SemidefiniteProgramming,” SIAM J. OPTIM., 18, No. 1, 186–205 (2007).

3. M. Vichi, G. Saporta. “Clustering and Disjoint Principal Component Analy-sis,” Computational Statistics and Data Analysis, (2008).

4. G.-W. Weber, P. Taylan, S. Ozogur, B. Akteke-Ozturk. “Statistical Learningand Optimization Methods in Data Mining,” in Ayhan, H. O. and Batmaz,I.: Recent Advances in Statistics, Turkish Statistical Institute Press, Ankara,2007, pp. 181–195.

5. R. Xu, D. Wunsch. “Survey of Clustering Algorithms,” IEEE Transactionson Neural Networks, 16, pp. 645–648 (2005).

6. The R Project for Statistical Computing, http://www.r-project.org/

169

On control with coefficients for high order partialdifferential equations

Igor E. Mikhailov1, L.A. Muravey2

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; mikh [email protected] MATI Russian State Technological University, Moscow, Russia; [email protected]

1. On classes of existence of optimization problems of formLet’s consider problem formulation on an example of form optimization

of a thin plate under the influence of the distributed loading described bythe model equation

∆(D(x, y)∆u) = q(x, y), (x, y) ∈ Ω ⊂ R2, (1.1)

where ∆ is the two-dimensional Laplace operator, 3√D(x, y) = h(x, y) is

the distribution of a thickness of the plate, u(x, y) is the form of the platedeflection , q(x, y) is the density of the distributed loading, the area Ωis the plate base. For definiteness we will consider the case of hingedfastening plate, with the following conditions on the boundary Γ = ∂Ω

u|Γ = ∆u|Γ = 0. (1.2)

It would appear reasonable that the equation (1.1) with boundaryconditions (1.2) is named as the state equation of plate L (it containsthe elliptic operator of the fourth order). The optimization problem is tominimize the integral functional

F (h) =

∫∫

Ω

f(x, y, u,∆u) dx dy, (1.3)

where the control parameter h (a thickness of the plate) belongs to someclass of functions K. An examples are the functional

F0(h) =

∫∫

Ω

[u(x, y)− z(x, y)]2 dx dy = ‖u− z‖2L2(Ω), (1.4)

expressing a deviation in norm L2(Ω) of the plate form u(x, y) with thick-ness h(x, y) from the set form z(x, y), or the functional

F1(h) =

∫∫

Ω

q(x, y)u(x, y) dx dy =

∫∫

Ω

D(x, y)(∆u)2 dx dy, (1.5)

170

expressing the work required for a deflection the plate of the thicknessh(x, y) under the influence of force with density q(x, y).

Let’s notice that the state equation L can be reduced to a system ofthe elliptic equations of the second order

∆u =v

D, (x, y) ∈ Ω, (1.6)

∆v = q, (x, y) ∈ Ω, (1.7)

u|Γ = v|Γ = 0. (1.8)

The equation (1.7) is a special case of the equilibrium equation of thefixed elastic membrane

div(k(x, y)∇v) = q, (x, y) ∈ Ω, (1.9)

v|Γ = 0, (1.10)

where function k(x, y), named by a tension factor, belongs to some classK which consists from measured in Ω functions k(x, y), satisfying almosteverywhere in Ω inequalities 0 < k0 ≤ k(x, y) ≤ k1 < ∞. Then atq ∈ L2(Ω) v(x, y) is the generalized solution of the problem (1.7), (1.8)

which belongs to the Sobolev spaceW 1

2(Ω).Let’s notice that for the analogue of the functional (1.5) F1(k) =∫∫

Ω

k(x, y)|∇v|2 dx dy the minimization problem in the class K has the

solution, that was shown by M. Goebel [1] (the Goebel’s method does notdepend on space dimension).

At the same time for functional F0(k) = ‖v−z‖L2(Ω) the minimizationproblem in the class K, named the Lions problem, has no solution (seecounter-examples of M. Murat [2] and D. Korsakova [3]). From theseexamples it follows that for resolvability of an optimization problem itis required to establish certain connection between the state equation L,minimized functional F and a class K of controlling functions, i.e. ofcoefficients in the equation (1.9). As was shown by E. De Giorgi, S.Spagnolo and T. Tartar [4], [5] for the solution of the Lions problem it isnecessary to replace the equation (1.9) with the equation

div(A(x, y)∇v) = q(x, y), (x, y) ∈ Ω,

171

where the symmetric positively defined matrix A(x, y), such that

0 < α|ξ|2 ≤ (Aξ, ξ) ≤ β|ξ|2, α < β, ξ ∈ R2,

belongs to the class K of controlling matrixes A if its own numbers α ≤λ1(x, y) ≤ λ2(x, y) ≤ β satisfy to relation λ1 = αβ/(α+ β − λ2).

Establishment of conditions on functional F (h) from (1.3), allowing tosolve considered minimization problems in the case of the state equationsof the fourth order in a class of scalar functions

K = k(x, y) : 0 ≤ h0 ≤ h(x, y) ≤ h1 <∞, (x, y) ∈ Ω , (1.11)

that is dictated by physical sense of the equation (h(x, y) is a plate thick-ness), has been made by L. Muravey, I. Ismailov, E. Eyniev, I. Mikhailovin [6], [7]. These conditions for the functional (1.3) are the continuitythe function f(x, y, t, τ) with respect to all of the variables, growth withrespect to variable t and convexity on variable τ .

These conditions, in particular, allowed to solve the minimizationproblem both for the functional F0(h), and for the functional F1(h), aswell as to develop effective numerical methods of construction of the ap-proximate solution of the optimization problem. As an example we willfulfill the calculation of a rectangular plate with the sides a = 1, b = 2.Restrictions in (1.12) were h0 = 0; h1 = 3. As z(x, y) in the functionalF0(h) was the function z(x, y) = 27

128x(a − x)y2(b − y). For this functionzy(x, 0) = 0.

Fig. 1

Fig. 1 shows the received distribution h(x, y). It is seen from Fig. 1that the thickness of a plate sharply increases near the line y = 0 andreaches the top restriction h1.

It confirms that the considered model adequately reflects the phe-nomenon, well-known from the elasticity theory, that at hinged fasteningof the loaded plate at y = 0 the maintenance of a zero deflection of a plateuy(x, 0) = 0 is possible only if its thickness is infinite in the vicinity y = 0.

Fig. 2 shows the plate deflection u(x, y).

172

Fig. 2

Thus the offered mathematical model and its numerical realizationqualitatively correctly describe a plate deflection.

2. Solution of the optimization problem in case of the generalequilibrium equation of a thin plate of a variable thickness

In this point as the state equation we will consider the equation [8]

∆(D(x, y)∆u) − (1− ν)(Dyyuxx − 2Dxyuxy +Dxxuyy) = q(x, y),(x, y) ∈ Ω ⊂ R2,

(2.1)with boundary conditions

u|Γ = ∆u|Γ = 0, (2.2)

where the Poisson number ν can vary in the limit from 0 to 1/2 (in practiceν =1/3, or ν =1/2 are taken).

The object of investigation is minimization of the functionals

F0(h) = ‖u− z‖2L2(Ω) (2.3)

F1(h) =

∫∫

Ω

q(x, y)u(x, y) dx dy =

=

∫∫

Ω

[D(∆u)2 − (1 − ν)(Dyyu

2x + 2Dxyuxuy +Dxxu

2y)]dx dy. (2.4)

Let’s notice that, as a plate is thin then the class K1 of controlling func-tions D(x, y) has the form

D(x, y) ∈ C2(Ω), 0 < α ≤ D(x, y) ≤ β <

∞, |Dxx|+ |Dxy|+ |Dyy| ≤ δ, where δ > 0 is sufficiently small. It is not

difficult to see that∣∣∣∣(1− ν)

∫∫

Ω

(Dyyu2x + 2Dxyuxuy +Dxxu

2y) dx dy

∣∣∣∣ ≤ (1− ν)δ‖∇u‖L2(Ω) ≤

173

≤ ε‖∇u‖L2(Ω) + C(ε, δ)‖u‖L2(Ω),

where ε is an arbitrary positive number, C(ε, δ) = const. It follows thatthe problem of the minimization of the functionals (2.3) and (2.4) can besolved in the class K1, a similar class K in item 1.

At the same time the aim of work was to obtain numerical estimatesof errors in the computation of the deformation and the thickness of theplates with the use of real equilibrium equation (2.1) in comparison withthe model equation (1.1). The calculations have shown, that upon mini-mizing the functional the real deformation of the plate described by theequation (2.1) with ν = 1/2, differed by more than 20% from the deforma-tion of a plate described by the equation (1.1). The difference of thicknessof plates was about 6%.

A separate work will be devoted to more detail study of the problemof minimization the functional F1(h).

References

1. M. Goebel. “Optimal control of coefficients in linear elliptic equations,” Mat.operations gorsch. undstatistics ser optimiz. 12, No. 4. 525–533 (1981).

2. M.F. Murat. “Un contre-example pour de problem du controle dans les coef-ficient,” CRAS ser A. Paris, 273, 708–711 (1971).

3. L.V.E. Korsakova. “Example of nonexistence of a solution of the Lions prob-lem on optimum control,” Probl. Mat. analysis LGU, 6, 60–67 (1977) (inRussia)

4. E. De Giorgi, S. Spagnolo. “Sulla convergenza degli integrali dell’energia peroperatori ellittici del secondo ordine,” Sezione Scientifica, Bollettino U.M.I.4, No. 8, 391–411 (1973).

5. L. Tartar. “Problemes de controle des coefficients dans les equations auxderives partielles,” Lect. Notes and Mat. Sus. 107, 420–426 (1975).

6. I. Ismailov, L. Muravey, E. Eyniev. “Some problems of structure optimiza-tion,” in: Works of the International Conference “Intellectual Systems. V. 1”,Leningrad, 1996, pp. 221–226 (in Russia).

7. I.E. Mikhailov, L.A. Muravey. “Analytical and numerical methods of solvingthe optimization problems of forms of mechanical structures,” in: Izbr. TrudyUniver. “Dubna”, No 1, Dubna, 2004, pp. 85–93 (in Russia).

8. V. Komkov. Optimal control theory for the damping of vibration of simpleelastic systems, Springer-Verlag, Berlin, Heidelberg, New-York, 1972.

174

Analytic design of an optimal controller underpermanent stochastic disturbances

Arsalan Mizhidon

East Siberia State university of Technology and Management, Ulan-ude,


In [1] there is a problem of analytic design of an optimal controllerunder permanent stochastic disturbances for the case of functionals withan integrand of the form x′Qx + u′Ru. In this paper the problem ofanalytic design of an optimal controller under permanent stochastic dis-turbances was generalized to the case of functionals with an integrand ofthe form x′Qx+ x′Pu+ u′Ru. The problem of analytic design of an op-timal controller under permanent determinate disturbances for such caseof functionals was considered in [2].

Consider a linear system with constant coefficients

x = Ax+Bu+Gz, (1)

where x — n-dimensional vector of state variables; u — r-dimensionalcontrol vector; A — (n × n) - matrix; B — (n × r) -matrix; z — p-dimensional vector of stochastic disturbances; G — (n× p) -matrix.

Suppose control constraint is not imposed. In addition, suppose thatz(t) is a persistent bounded stochastic function of time, which can berepresented as the solution of a linear system of stochastic equations

z = Dz + ξ(t). (2)

Here D — p× p - matrix, ξ(t) — vector stochastic process as white noise.Stochastic process modeling, as white noise transmission through lin-

ear system (2), matches to stochastic stationary process modeling with aknown spectral energy density. Obviously matrix D is stable, otherwisethe solution of the equation (2) would be an unlimited stochastic process.

It is required to define control u(x, z) minimizing following functional

J(u) = limT→∞

1

2TM

T∫

0

(x′Qx+ x′Pu+ u′Ru)dt, (3)

175

where Q — nonnegative-defined constant n × n - matrix; P — constantn × r - matrix; R — positive-defined constant r × r - matrix; M(.) —mathematical expectation. In addition, suppose matrices Q, P , R satisfythe requirement: matrix W , consisting of these matrices,

W =

(Q 1

2P12P

′ R

),

is positive-defined matrix.Let us introduce expanded state vector y = (x, z)′. Because of (1), (2)

this vector satisfies the system of differential equations

y = Ay + Bu+ f(t). (4)

Here

A =

(A G0 D

), B =

(B0

), f(t) =

(0ξ(t)

).

Functional (3) will become

J(u) = limT→∞

1

2TM

T∫

0

(y′Qy + y′P u+ u′Ru)dt, (5)

where

Q =

(Q 00 0

), P =

(P 00 0

).

It was shown that the solution of stochastic optimal control problem ofsystem (4) on a finite interval of observation with the performance measure

J(u) =1

2M

T∫

0

(y′Qy + y′Pu+ u′Ru)dt, (6)

results in the following.The presence of white noise ξ(t) in the system (4) does not change

the solution of determinate problem of designing an optimal controller[2], except for increase in minimum value of performance measure. Thus,optimal control in problem with performance measure (6) is given by

u∗(y, t, T ) = −(R−1B′K(t, T ) +

1

2R−1P ′

)y,

176

where K(t, T ) — solution of matrix differential equation

K +KA+ A′K −KBR−1B′K − 12KBR

−1P ′ − 12 PR

−1B′K−− 1

4 PR−1P ′ + Q = 0,

(7)

with a condition on the right end

K(T ) = 0.

Based on research of solutions K(t, T ) of matrix differential equation (7)with a zero condition on the right end as T → ∞ there were obtainedconditions of existence of following limit

limT→∞

K(t, T ) = K,

where K — constant symmetric positive-defined matrix and K is a solu-tion of matrix algebraic equation

KA+A′K−KBR−1B′K−1

2KBR−1P ′−1

2PR−1B′K−1

4PR−1P ′+Q = 0.

It was proved that optimal control in problem (4)-(5) is given by

u∗(y, t) = −(R−1B′K +

1

2R−1P ′

)y.


(project no. 12-08-00309).

References

1. A.D. Mizhidon “Analytical design of optimal controllers under permanentstochastic disturbances as applied to the design of vibration isolation sys-tems”, Automation and Remote Control, No. 4, 81–93 (2008).

2. A.D. Mizhidon “On a problem of analytic design of an optimal controller”,Automation and Remote Control, No. 11, 102–116 (2011).

177

Optimal control of the sweeping process

Boris Mordukhovich1

1Wayne State University, Detroit, USA; [email protected]

We formulate and study an optimal control problem for the sweeping(Moreau) process, where control functions enter the moving sweeping set.To the best of our knowledge, this is the first study in the literature de-voted to optimal control of the sweeping process. We first establish anexistence theorem of optimal solutions and then derive necessary optimal-ity conditions for this optimal control problem of a new type, where thedynamics is governed by discontinuous differential inclusions with vari-able right-hand sides. Our approach to necessary optimality conditionsis based on the method of discrete approximations and advanced toolsof variational analysis and generalized differentiation. The final resultsobtained are given in terms of the initial data of the controlled sweepingprocess and are illustrated by nontrivial examples.

This talk is based on the joint work with G. Colombo, R. Henrion,and N. Hoang

Optimal deformation during the creep

Evgenii Murashkin1

1 Institute for Automation and Control Processes Far Eastern Branch Russian

Academy of Sciences, Vladivostok, Russia; [email protected]

In the design of ship hulls and aircraft are widely used panels and pro-files of hardly-deformed aluminum alloys at normal temperatures. Tra-ditional methods of formation of structural elements of such alloys oftenleads to the appearance of the plastic breaks, cracks and other damage.In this regard, effective way is formation under high temperature and lowdeformation under creep conditions. The use of such process ensures theproduction of parts with high accuracy, which reduces the complexity of

178

assembly and welding, and also reduces the formation of effort to improvethe residual service life and quality of construction. At the same time itis necessary to solve problems of modeling the process of formation, thetiming and strength of deformation regimes, the determination of the ge-ometry of the light snap of an elastic response after removal of the loadingeffort. In this case there is a necessity of calculation of such processes inthe model with finite irreversible deformations and complicated rheologi-cal properties of materials. Consideration be carried out in the model offinite elastoplastic deformations [1, 3], the basic kinematic relations onein the Cartesian system (Eulerian coordinates) can be written in the formof

DeijDt

= εij − γij −1

2((εik − γik + zik)ekj + eik(γkj − εkj − zkj)) ,

DpijDt

= γij − pikγkj − γikpkj , (1)

DnijDt

=dnijdt− riknkj + nikrkj , rij = wij + zij (eij , εij) ,

εij =1

2(vi,j + vj,i) , wij =

1

2(vi,j − vj,i) ,

dij =1

2(ui,j + uj,i − ui,kuk,j) =

eij + pij −1

2eikekj − eikpkj − pikekj + eikpkmemj.

Here ui, vi – components of displacement and velocity vectors; eeij =eij−0.5eikekj and pij – reversible and nonreversible parts of total Almansistrains; D/Dt – an objective derivative; source γij in the equation oftensor transfer, is a tensor of velocities of nonreversible deformations, zij– non-linear part of rotation tensor rij , completely written out in [2],defining its difference from a rigid rotation tensor wij . This assumptionallows us to write the stress-strain relations like Murnagan’s relations intheory of non-linear elasticity.

σij = −pδij +∂W

∂eik(δkj − ekj) ,

W = (α− µ)J1 + αJ2 + βJ21 − κJ1J2 − ζJ2

1 ,

179

J1 = eejj , J2 = eeijeeji,

whereW (J1, J2) is a elastic potential and α, µ, β, κ, ζ are elastic constants.In the course of deforming, anticipating plastic flow, and in the course

of unloading, source γij in the equation of tensor transfer (1) for tensorof nonreversible strains pij is identified with a velocity of creep strainsγij = εvij for which we accept the power law (Norton-Bailey law)

εvij =∂V (Σ)

∂σij, V (Σ) = BΣn (σij) ,

Σ =

√3

2((σ1 − σ)2 + (σ2 − σ)2 + (σ3 − σ)2).

Here σ1, σ2, σ3 are principal values of Cauchy stress and B, n are creepparameters.

We formulate the optimal deformation problem in the creep: we arenecessary to define a way of deformation of elasto-creeping material withina specified period of time to the final point of unloading to obtain thespecified values of residual strains with minimal damage parameter. Thusthe problem reduces to a multi-criteria optimization problem [4,5]

J0 = w1

(∫

V

(A(ε, σ, γ) +A(ε∗, σ∗, γ))dV −∫

S

piuidS

)+w2

(∫

V

σijγijdV

),

A(ε, σ, γ) =1

2(σijεij − σijγij).

Where w1, w2 are weights, pi are boundary loads, ε∗, σ∗ are residualstrains and stresses. The proposed model can be applied to process opti-mization in terms of reduction of residual stress and to identify the mech-anism of ”healing” of microdefects at improving performance propertiesof the finished product.

The authors were supported by the Grant of the President of the Russian

Federation (project no. MK-776.2012.1).

References

1. A.A. Burenin, L.V. Kovtanyuk, M.V. Polonik. The formation of a one-dimensional residual stress field in the neighbourhood of a cylindrical defectin the continuity of an elastoplastic medium. Journal of Applied Mathematicsand Mechanics 67, 283–292, 2003.

180

2. A.A. Burenin, L.V. Kovtanyuk, M.V. Polonik. The possibility of reiteratedplastic flow at the overall unloading of an elastoplastic medium. DokladyPhysics 375, 767–769, 2000.

3. A.A. Burenin, L.V. Kovtanyuk, E.V. Murashkin. On the Residual Stresses inthe Vicinity of a Cylindrical Discontinuity in a Viscoelastoplastic Material.Journal of Applied Mechanics and Technical Physics 47, 241–248, 2006.

4. I.Yu. Tsvelodub. A class of inverse creep theory problems. Journal of AppliedMechanics and Technical Physics 30, 320–329, 1989.

5. K. Washizu. Variational Methods in Elasticity and Plasticity. 3rd edn, Perg-amon Press, London, 1982.

Modeling and optimization of ion-beam etchingprocess

L.A. Muravey1, V.M. Petrov1, A.M. Romanenkov1

1 MATI–RSTU, Moscow, Russia; [email protected]

Introduction. The problem of optimal controlling the ion-beam etch-ing (IBE) process with the purpose of minimizing the geometrical sizesof the elements being etched is solved by changing the angle of incidenceof the ion beam with respect to the target. The evolution of the surfacehaving an arbitrary form in the ion bombardment process is described bythe equation:

zt(t, x) + ν(θ)√

1 + z2x(t, x) = 0, (1)

where z(t, x) is the height of the etched form at the time moment t inx position, ν(θ) is the material etching velocity, depending on the angle

Fig. 1 Fig. 2

181

θ, formed by the angle of incidence and the normal to the surface beingetched. In particular, if the ion beam is perpendicular to axis x, thenθ = arctg ∂z

∂x(t, x). The specific feature of all existing resists is the non-monotone nature of function ν(θ), i.e. the existence of a certain θ = θ∗, atwhich ν(θ) is maximum. As a rule, the function ν(θ) is determined exper-imentally and normally has the form illustrated in Fig. 1. The directionof the ion beam can vary with time, then the angle θ can be presented inthe following form:

θ = arctg∂z

∂x+ α(t), (2)

where α(t) is the angle between the direction of the ion beam incidenceand the axis y. α(t) will be considered as the control, upon which thenatural restrictions have been imposed (see Fig 2).

0 ≤ α(t) ≤ αmax. (3)

IBE equation derivation. The ion-beam etching equation will beobtained in the assumption that during the etching process the sample isrotating with the angle velocity ω with respect to the vertical straight line,passing through the semicircle centre. It should be noted that the two-dimensional and three-dimensional cases differ only by form for θ angle,and the evolution process of the being etched profile is described by thesame equation. Also, it is possible to show, that when ω = 0, then weagain obtain the two-dimensional case.

Let z = z(x, t) to be a smooth function, describing the etched surfaceevolution, Pt = x, z(x, t) is the etching profile. Let’s consider the pointQ ∈ Pt, we will draw the nQ normal vector from the given point:

nQ =1√

1 + z2x(x, t)−zx(x, t), 0, 1. (4)

As the mask rotates with the angular velocity ω, then (see Fig. 3) wehave

γ = sinα sinωt, sinα cosωt, cosαNote, that |nQ| = |γ| = 1, therefore (see Fig. 2) (nQ, γ) = cos θ. That

is

cos θ =cosα− zx(x, t) sinα sinωt√

1 + z2x(x, t). (5)

182

Through the time period ∆t the point Q will move to the pointM(x1, z1) on the profile Pt+∆t. This movement will be executed alongthe vector −nQ with νQ velocity. Let’s consider the vector QM = x1 −x, z1 − z. Based on the etching physical law it can be stated that thevector QM can be determined by the following formula:

QM = −ν(θ)∆tnQ = −∆t ν(θ)√1 + z2x(x, t)

−zx(x, t), 0, 1.

Using the theorem of the finite increments we will obtain:

−∆t ν(θ)√1 + z2x(x, t)

= ∆tν(θ)√

1 + z2x(x, t)z2x(x, t) + zt(x, t)∆t+ o(∆t).

After division by ∆t, and directing it to 0 and after the elementary trans-formations we have equation (1).

On the evolution of the smooth convex profile. Effect offracture appearance. Note that if the mask initial form is smooth andconvex, then in its evolution process the appearance of the angular pointsis possible. Let’s consider this effect in more detail.

Let us consider a simple example α = 0, −→γ ‖ Oz and the initial profilein the form of a semicircle

z |t=0=√1− x2, | x |≤ 1, (6)

where −→γ -ion beam direction. In this case the solution of problem (7) canbe represented in the analytical form. Let us give the unit normal vectorprofileSt in the form ψ = sin θ, cos θ, |θ| ≤ Π

2 and the initial profile S0

in the form S0 = (x, z) : x = sin θ, z = cos θ, |θ| ≤ Π2 and introduce the

functionsF (θ) = v(θ) sin θ + v′(θ) cos θG(θ) = v(θ) cos θ − v′(θ) sin θ

. (7)

Theorem 1. Then there exists the time T > 0, for which on theinterval 0 < t < T the profile St will remain smooth convex and will berepresented as

St = (x, z) : x(θ, t) = sin θ − tF (θ), z(θ, t) = cos θ − tG(θ), |θ| ≤ Π/2.(8)

183

Fig. 3 Fig. 4

So, the time T may be found from relations

∂x

∂θ= cos θ − t cos θ(v(θ) + v′′(θ)) = 0,

∂z

∂θ= − sin θ + t sin θ(v(θ) + v′′(θ)) = 0,

i.e. 1/T = maxR(θ), |θ| ≤ Π2 , where R(θ) = ν(θ) + ν′′(θ). In our test

example ν(θ) = (1+2 sin2 θ) cos θ we have R(θ) = 4 cos θ(4 cos2 θ−3) andmaxR(θ) = R(Π/3) = 4, hence T = 1/4 (see Fig. 4).

Consider the case of non-smooth convex profile.

C(St, ψ) = max(xψ1 + zψ2), (x, z) ∈ St. (9)

Let us consider the gradient

C′(S0, ψ) =

[∂C

∂ψ1(S0, ψ),

∂C

∂ψ2(S0, ψ)

](10)

and give [u(θ), v(θ)] = C′(S0, ψ)|ψ=(sin θ,cos θ)

Every non-smooth initial profile can be approximated be means of asmooth one according to the following procedure. Let us consider therectangular profile Π = |x| ≤ a, |y| ≤ b with the support functionC(Π, ψ) = a|ψ1| + b|ψ2|. The smoothing profile Πµ is determined bymeans of the support function

C(Πµ, ψ) =√a2ψ2

1 + µ2ψ22 +

√µ2ψ2

1 + b2ψ22

184

Fig. 5 Fig. 6

where µ > 0 is a small parameter. Then due to (9), (10) we have

Sµ,t = (x, z) : x = uµ(θ) − tF (θ), z = vµ(θ)− tG(θ), |θ| ≤ π/2

(see Fig. 5).One of the methods to avoid distortion of the geometrical sizes (see

Fig. 6) is the possibility to change the ion beam direction in the etchingprocess, i.e. of the angle change in time. The equation for this velocityof x0(t) (see Fig. 6) can be easily derived from the geometrical consider-ations. Namely

dx0dt

= ν

(α+ arctg

∂z

∂x

∣∣∣z=0

)√1 +

(∂z∂x

)2∣∣z=0(

∂z∂x

)∣∣z=0

(11)

So, the optimal control problem consists in the following. To determinethe functions z(t, x) and a(t), satisfying the equations (1), (11) the re-strictions (3), the initial conditions

z(0, x) = g(x), (12)

where g(x) is the convex function, and the condition

z(T, 1) = −H. (13)

In this connection it is necessary that at the finite moment of time t = Tthe function x0(t) should take the maximum value, i.e.

J = x0(T )→ max . (14)

Note that the formulated problem is the problem with the non-fixedtime T . This creates some additional difficulties, because in addition to

185

the functions α(t) and z(t, x) also, it is necessary to look for the processtermination moment T . In our case this difficulty can be easily avoided.For this the function h(t) – the depth of the working layer current etch-ing – will be introduced. This function is monotone and varies within0 ≤ τ = h(t) ≤ H limits. Then the equations (1) and (11) will have thefollowing form

∂z(τ, x)

∂τ+v(θ + α)

v(α)

√

1 +

(∂z

∂x(τ, x)

)2

= 0, (15)

dx0dτ

=v(θ + α)

v(a)

√1 + ( ∂z∂x )

2|z=0

( ∂z∂x)|z=0

, (16)

with the conditions (12) and

x0(0) = 1, 0 ≤ x ≤ 1, 0 ≤ τ ≤ H. (17)

So, our problem of the optimal control has been reduced to the search forthe functions α(τ) and z(τ, x) satisfying the equations (15), (16), condi-tions (17) and providing the maximum (14) with restriction (3).

Solution method. If we consider numerical solutions z(τj , xi) andx0(τj), xi = i/m, i = 0,m, τj = Hj/n, j = 0, n of problem (15)–(17), thendue to Pontryagin maximum principle for every τj we can see the optimalvalue αj of control function αj as min

kv(θj +αk)/v(αk), 1 ≤ k ≤ n, where

θj = arctg ∂z/∂xi∣∣(z=0,x0(τj)=i/m)

.

186

Special algorithm for Three-Stations Railway problem

Elena Musatova1, Alexander Lazarev2, Nail Husnullin3

1 Institute of Control Sciences RAS, Moscow, Russia; [email protected] Institute of Control Sciences RAS, Moscow State University,National Research University – Higher School of Economics,

Moscow, Russia; [email protected] Institute of Control Sciences RAS, Moscow, Russia; [email protected]

Problem statement

The paper is devoted to the problem of railway transportation. Therailway network consists of stations between which freight cars are trans-ported. Let us S be a set of stations. Every station s has a setNs = J1, ...Jns

of orders to deliver. N = ∪s∈SNs is a common setof orders. Each order represents one freight car. If an order consists ofk cars, we will consider it as k different orders. We know realise time rsjand due date dsj of delivery for each car Jj ∈ Ns. Let us psj be traversingtime for the car Jj ∈ Ns and wsj be its weight (importance). Our goal isto design freight trains and work out their schedule. Objective functionscan be the following:

• minimizing the weight total tardiness

min∑

s∈S

∑

j∈Ns

wsj max0, Csj − dsj;

• minimizing the total completion time

min∑

s∈S

∑

j∈Ns

Csj ;

• minimizing the maximal lateness

min maxj∈Ns,s∈S

Csj − dsj;

• in the set N find subset N ⊆ N of cars that can be delivered ontime:

max∑

j∈N

wsj −∑

j∈N\N

zsj .

187

These problems are large-scale and difficult to solve. Therefore, weproposed to divide them into subproblems which are easier to solve andconsider special cases which would help to find important structural prop-erties which are hard to recognize in the general case. We have suggested anumber of railway basic models (with two stations, with chain of stationsand so on) that gives us an opportunity to develop special exact algo-rithms which can be used in general railway problems. Some algorithmsfor two-stations railway problems can be found in [1]. In this paper wepropose an algorithm for the special case of three stations.

Three-Stations Railway problem

Consider the problem with 3 stations that are connected by a railroad and one locomotive. A valid arrangement is shown on fig. 1. Arrowsindicate a possible route of the locomotive from one station to another.

Fig. 1 Possible arrangement of stations

We have to implement a set of orders. N ij is a set of orders that shouldbe delivered from the station i to the station j. So N1 = N12 ∪N13 etc.Let us assume that p is a traversing time from one station to another, qis a capacity of a train, rsi is a release time of Jsi ∈ Ns, s ∈ 1, 2, 3, ns isa common number of orders at the station s, nsi is a number of orders atstation S that are should be delivered to the station i.

Objective function of the problem is following:

min∑

J1i ∈N1

C1i +

∑

J2i ∈N2

C2i +

∑

J3i ∈N3

C3i . (1)

It is easy to see the locomotive has the following strategies when hearrives to a station:

1. staying at the station and waiting a new order;

188

2. idling to the next station;

3. idling to the previous station;

4. moving to the next station with the largest possible number of cars;

5. moving to the previous station with the largest possible number ofcars.

Only for this objective function (1) we can reduce the number of pos-sible solutions if we consider the following:

• idling to the next or previous station is the same as moving withoutthe wagons;

• if we can move at the second station with q wagons it is more prefer-able for us than stay at the station;

• idling to another station is preferable, if all orders have been deliv-ered from station s.

It is obviously that in an optimal schedule the train begins his move-ment from a station s only at the moments of its arrival to this stationor at the moment of appearance of a new order, i.e. at the moment rsi .So times points at which the train begins and ends movement betweenstations belong to T = t : ∃rsj , ∃l ∈ 1, . . . , (n1+n2+n3), t = rsj + lp.

Let us denote by

S(s, t, k12, k13, k21, k23, k31, k32) (2)

the state at the moment t ∈ T , where s is the number of the station wherethe locomotive is, k12 is the number of delivered orders from the first to thesecond station, k23 the number of delivered orders from the second stationto the third one, etc. Let us assume that P (s, t, k12, k13, k21, k23, k31, k32)is the smallest total delivery time in the scheduling which leads to stateS(s, t, k12, k13, k21, k23, k31, k32). For the objective function (1) the opti-mal solution of the problem is

mins,t

P (s, t, n12, n13, n21, n23, n31, n32). (3)

E x a m p l e 1. Consider the following problem. Let us assume thatr12 = (1, 2, 3), r23 = (2, 3, 4), r13 = (3, 4, 5), r21 = (2, 3), r32 = (1, 2),

189

r31 = (2, 3), p = 2, q = 2. One of the possible scheduling solutions isshown on fig. 2.

Fig. 2 One of the possible scheduling solutions

Algorithm 1 describes the calculation of the number of cars that thetrain can take to the next station in (2).

Algorithm 1

1: function GetPossibleOrders(s,ns,futureS)2: futureQ← 03: while rsi < t do

4: if i > n & i− n <= q then futureQ← futureQ+ 15: end if

6: end whilereturn futureQ

7: end function

Let us introduce the following denotations:t — current time;s — station number, where the locomotive is located at the time t;j — number of wagons, which the locomotive can take at the currenttime;n — number of delivered orders from station s;q — max number of wagons that the locomotive can carry at a time;N [s][futureS] — array, which contain number of delivered orders fromstation s to futureS;Runner — entry point, which execution of the program begins with;existsCarsOnStay— function, which returns false, if all orders have beendelivered from station s.

190

Algorithm 2 creates nodes of the tree and allows to move from onestation to another.

Algorithm 2

1: function Runner

2: int[,] N ← new int[s,s]3: N ← 04: newS ← S(1, 0, N)5: BuildTree(newS)6: end function

7:

8: function BuildTree(prevS)9: j ← 1

10: q ← 011: s← prevS.s

12: t← prevS.t

13: N ← prevS.N

14: while j..3 do

15: if j = 1 then futureS ← s⊕3 116: end if

17: if j = 2 then futureS ← s⊖3 118: end if

19: if j = 3 then futureS ← s

20: end if

21: q ← GetPossibleOrders(s,N [s, futureS])22: N [s, futureS]← N [s, futureS] + q

23: if j <> 3 then

24: t← t + p

25: else

26: t← t + 127: end if

28: newS ← S(futureS, t, N)29: if existsCarsOnStay(newS) then

30: BuildTree(newS)31: end if

32: end while

33: end function

As we have O((n1 + n2 + n3)2) possible time moments running time of

the proposed algorithm is O((n1 + n2 + n3)2n12n13n21n23n31n32).


(project no. 11-08-13121).

References

1. A.A. Lazarev, E.G. Musatova, E.R. Gafarov, A.G. Kvaratskhelia. Scheduletheory. Problems of railway planning, ICS RAS, Moscow (2012).

191

Numerical solution of two-dimensional inverseproblem for the Helmgoltz equation

Daniar Nurseitov1, Maksim Shishlenin2, Syrym Kasenov3

1 K.Satpaev KazNTU Information and Space Technologies National Scientific

Laboratory, Almaty, Kazakhstan; [email protected] Sobolev Institute of Mathematics of the Siberian Branch of the Russian

Academy of Sciences, Novosibirsk, Russia; [email protected] Institute for Master and PhD programs Kazakh National Pedagogical

University named after Abai, Almaty, Kazakhstan; [email protected]

We consider the Helmholtz equation in Ω = (−b, b) × (0, L) initial-bondary problem:

uzz + uyy − r2(y, z)u = 0 (y, z) ∈ Ω (1)

uy(−b, z) = 0, uy(b, z) = 0, z ∈ [0, L] (2)

uz(y, 0) = g(ω, y), uz(y, L) = 0, y ∈ [−b, b] (3)

here r2(y, z) = ω2

υ2(y,z) υ−the velocity of wave propagation; Inverse

Problem: find the coefficients υ(y, z) of equation (1) using additional in-formation about the solution to the initial-boundary problem (1)-(3):

u(y, 0, ω) = f(y, ω), y ∈ [−b, b] (4)

This problem (1)-(4) is ill-posed. We write the problem (1)-(4) in theoperator form Aυ = f . We solve numerically the equation Aυ = f , weminimize the objective functional

J(υ) = 〈Aυ − f,Aυ − f〉 =∑

ω

b∫

−b

[u(y, 0, ω; υ)− f(y, ω)

]2dy (5)

We minimize the quadratic functional (5) using the steepest descent method[1]:

υn+1 = υn − αnJ ′(υn) n = 0, 1, . . . , (6)

where υ0 is an initial approximation, J ′(υn) is the gradient of the objectivefunctional, and the descent parameter αn is determined from the condition

192

αn = argminαJ(υn − αJ ′(υn)).

Let us consider adjoint problem:

ψzz + ψyy − r2(y, z)ψ = 0 (y, z) ∈ Ω (7)

ψy(−b, z) = 0, ψy(b, z) = 0, z ∈ [0, L] (8)

ψz(y, 0) = 2(u(y, 0, ω)− f(y, ω)

), ψz(y, L) = 0, y ∈ [−b, b] (9)

Gradient of functional:

J(υ) =∑

ω

ω2

υ4(y, z)u(y, z, ω)ψ(y, z, ω) (10)

Algorithm for solving the inverse problem is 1. Choose an initialapproximation υ0;2. Solve the direct problem (1) (3) with we get u(y, 0, ω);3. We calculate the boundary condition of adjoint problem (9);4. Solve the adjoint problem (11)-(13) and we get ψ(y, z, ω);5. Find the gradient by the formula (10);6. We calculate the υ1 by the formula (6);7. We verify value of the functional(5), when he reached the minimum,then the problem is solved, otherwise the return to step 1. using theυ0 = υ1.

References

1. S.I. Kabanikhin Inverse and Ill-Posed Problems: Theory and Applications,De Gruyter, Germany (2011)pp.459

2. S.I. Kabanikhin, M.A. Bektemesov, A.T. Nurseitova Iterative methods forsolving Inverse and Ill-Posed problems with the data on the part of the bound-ary, PF International Fund for inverse problems, Almaty, Novosibirsk, 2006,pp.315. (in Russian)

193

Production model in the conditions of unstabledemand

Nataliya Obrosova1, Alexander Shananin2

1 CC RAS, Moscow, Russia; [email protected] MIPT, Moscow, Russia, CC RAS, Moscow, Russia; [email protected]

Russia’s accession to the World Trade Organization aggravated prob-lems of product competitiveness of the Russian production as within thecountry, and in the world market. Thus industries of the Russian econ-omy appear in various conditions. Domestic raw materials corporationswere included long ago into the world energy market. Products of an oiland gas complex since the Soviet period are competitive. More difficultis a situation in processing industries. Historically production capacitiesof this sector were created in the conditions of the closed economy andabsence of the competition to the import goods. Up to now there is a tech-nological backwardness of processing sector remained. Products of sectorlose in the competition to better import analogs. As a result the producerhas delays with sales of products and current assets deficit which becomescovered or at the expense of bank loans, or at the expense of the stategrants. The inefficiency of processing sector essentially influences eco-nomic indicators of production, and also economy indicators as a whole.The program implementation of upgrade of processing sector shall be car-ried out after the detailed analysis of their economic consequences. Suchanalysis shall be carried out taking into account feedback and potentialinfluence of change of indicators of an industry on state of the economyas a whole. The adequate tool for carrying out such researches are themathematical models of economy constructed on the basis of a systemapproach [1] to the analysis of economic events and allowing analyzing aconsequence of large economic decisions taking into account their indirectconsequences.

In 2005-2007 based on a system approach the model of economy ofRussia intended for the purposes of short-term and mid-term forecasting[2] was developed. In the model the description of activities of processingsector taking into account the current assets deficit developed in [3] for thefirst time was used. The calculations for model showed [2] that withoutan inefficiency of processing sector growth rate of economy appears over-

194

estimated by 2-3 %. Thus features of processing sector need additionaldetailed research on the basis of adequate mathematical models. Resultsof this research are provided in the report.

Let’s assume that demand for a made product is unstable. Sale ofa product occurs during the random moments of time forming a Poissonprocess with parameter λ. As a result the producer is forced to accumulatesome amount of products in a warehouse in hope of sale. If sale doesn’tcome the producer has a current assets deficit which it is possible to coverat the expense of the short-term credit line K(t) under percent r.

Let’s designate τ the maximum time which is profitable to the pro-ducer to use the credit line K(t) in the conditions of absence of sale. LetY ∗- restriction of trade infrastructure which is understood as the greatestpossible consignments. Let’s designate: Y0 - the current product stock ina warehouse of the producer, η - production capacity, y - cost value of aproduct, p - the product price. Then

K (t) =

yη, 0 ≤ t ≤ τ0, t > τ

, (1)

the loan debt L(t) changes owing to the equation

dL(t)dt = K (t) + rL(t)

L(0) = 0, i.e. L(t) =

yη

r

(ert − er(t−τ)+

). (2)

The product output by the time t is

Y (t) =

Y0 + ηt, 0 ≤ t ≤ τY0 + ητ, t > τ

. (3)

The purpose of the owner of manufacture is maximization of the in-comeW (Y0) discounted with coefficient ∆ ≥ r on the unrestricted horizonchoosing time τ0 of credit using. The discounted income of the owner ofmanufacture W (Y0) is the solution of the following Bellman equation

W (Y0) = maxτ≥0

∞∫0

λe−(λ+∆)t[pmin (Y (t) , Y ∗)− L (t) +W

((Y (t)− Y ∗)+

)]dt.

(4)

From the economic point of view W (Y0) characterizes firm cost in case ofY0 a product stock.

195

Theorem 1. The solution W (Y0) of the equation (4) exists and isunique in a class G[0,+∞) of the continuous, non-negative, not decreas-ing, concave functions limited together with the derivative on a semi-interval [0,+∞), i.e.

G[0,+∞) =

w(x)| x ∈ [0,+∞), w ∈ C[0,+∞), 0 ≤ w(x) ≤ λ∆pY

∗,0 ≤ dw

dx ≤ p, w (αx + (1− α)y) ≥ αw(x) + (1− α)w(y)∀x, y ∈ [0,+∞), α ∈ [0, 1]

.

If the first derivative in some point for function w isn’t determined(owing to monotony the first derivative can have only discontinuity of thefirst kind and a set of discontinuity points is not more than countable),as a derivative we will understand a function derivative on the left in allpoints, except 0, and in 0 - a function derivative on the right.

It is proved that the solution of the equation (4) satisfies the condition

W (0) ≤ η

∆

λ

λ+∆

(p− y (λ+∆)

λ+∆− r

)

+

.

We suppose everywhere further that the condition of profitability of pro-

duction p− y(λ+∆)λ+∆−r > 0 is fulfilled.

The equation (4) was reduced to the integral equation that allowed toprove the following theorem.

Theorem 2. The optimal period for using credit τ0 =(τ02 + Y ∗−Y0

η

)+,

where τ02 = argmaxτ ≥0

τ∫0

[W ′ (ηt)− y(λ+∆)

λ+∆−r

]+e−(λ+∆)tdt.

Denote

ς0 =τ02 ηY ∗ , Y0 = ςY ∗, χ = (λ+∆) Y

∗

η , β =λ

λ+∆p−y(λ+∆)λ+∆−r

λλ+∆(p−

y(λ+∆)λ+∆−r )

,

Ψ0 =(

∆ηW (0)− λ

λ+∆

(p− y(λ+∆)

λ+∆−r

))(λ

λ+∆

(p− y(λ+∆)

λ+∆−r

))−1

.

Parameter 0 < β ≤ 1 is connected to the profitability of production. Inforce of restriction on W (0) the value Ψ0 ∈ [−1, 0] .

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Theorem 3. The function W (Y0) ∈ G[0,+∞) is the solution of theequation (4) if and only if the function

Φ (ς) =(λ

λ+∆

(p− y(λ+∆)

λ+∆−r

))−1

e−(λ+∆)ςY ∗/η[W ′(ςY ∗)− y(λ+∆)

(λ+∆−r)

] (5)

is the solution of the system

Φ (ς) =

βe−χς +Ψ0, if 0 ≤ ς ≤ 1,

Ψ0 + e−χ − (1− β) e−χς − λλ+∆χe

−χς−1∫0

Φ+ (ξ) dξ +

λλ+∆ Φ− (ς − 1) , if ς > 1,

(6)

where ς0,Ψ0 is the solution of the system

Φ (ς0) = 0,

Ψ0 + e−χ = λλ+∆χe

−χς0∫0

Φ (ξ) dξ.(7)

If the system (7) doesn’t have solution, then ς0 = 0, Ψ0 = −e−χ.

Corollary 1. Let W (Y0) ∈ G [0,+∞) is the solution of (4). Then thefollowing statements are true.

1. If profitability condition taking into account sales volume restrictionβ > e−χ is true, then

τ02 = argmaxτ ≥0

τ∫

0

[W ′ (ηt)− y(λ+∆)

λ+∆− r

]

+

e−(λ+∆)tdt > 0.

2. If β ≤ e−χ, then τ02 = 0, i.e. manufacture works only before achieve-ment of a stock Y ∗.

By the help of steps method the solution of the system (6) is obtainedin an explicit form [4].

Corollary 2. For 0 < ς0 ≤ 1 it is necessary and sufficient that

e−χ < β ≤ β1, where β1 =(1 + λ

λ+∆ − λλ+∆ (1 + χ) e−χ

)−1

.

The stock of product in the warehouse at the moment of time t isς(t)Y ∗. Change of value ς(t) is a random process. The analysis of theprocess allowed finding average loading of manufacture.

197

Theorem 4. In the case of ς0 ∈ [0, 1) average loading of manufactureis u0 = 1− exp (−λς0Y ∗/η∗).

The developed model of production can be used for the description offeatures of functioning of processing sector in model of Russia economy.


(projects N11-07-00162, 11-01-12084-ofi-m, PFI OMN RAN N3 (project N3.14),

PFI of Presidium RAN N15 (project N106).

References

1. A.A. Petrov, I.G. Pospelov, A.A. Shananin. Opit matematicheskogomodelirovaniya economiki. Energoatomizdat, Moscow ( 1996).

2. N.K. Obrosova, A.V. Rudeva, A.Yu. Flerova, A.A. Shananin. Ocenkavliyaniya gosudarstvennoj energeticheskoy politiki na perehodniye processi veconomike Rossii, CC RAS, Moscow (2007).

3. A.V. Akparova, A.A. Shananin. “Model proisvodstva v usloviyahnesovershennoy kreditnoy sistemi i nestabilnoy realizacii produkcii”,Matematicheskoye modelirovaniye, 17, N9, 60-76 (2005)

4. N.K. Obrosova, A.A. Shananin. “Kvantoviy effect v modeli proisvodstva suchetom deficit oborotnih sredstv i torgovoy infrastrukturi”, Trudi institutamatematiki i mekhaniki UrO RAN, 16, N5, 127-134 (2010).

Global search in bilinear separation problems

Andrei Orlov1, Sergei Pinigin2

1 Institute for System Dynamics and Control Theory, Siberian Branch of the

Russian Academy of Sciences, Irkutsk, Russia; [email protected] Institute for System Dynamics and Control Theory, Siberian Branch of the

Russian Academy of Sciences, Irkutsk, Russia; [email protected]

1. Introduction. The work is devoted to one of the problems of theso-called generalized separability [1, 2]. This problem can be arisen inmany practical areas, where we need procedures for classifying objects.

198

The practical applications concern, for example, technical problems (di-agnostic materials and nanomaterials), economic problems (banks andcredits), medical problems (diagnosis and prognosis), and so on.

The purpose of this procedure is the classification of objects by in-vestigation of available statistic data to certain classes by constructing aspecial rule (discriminant function). The simplest discriminant function islinear function [1]. But, as known, sets are linearly separable if and onlyif the intersection of their convex envelopes is empty. So, the practicaldemands are not limited by the linear separability only, and one oftenneeds a more general concept of separability, e.g. bilinear separability [2].

2. Problem formulation and its reduction. Consider two non-empty finite sets A and B from the space IRn. These sets include m and kpoints, respectively, so that A and B can be represented by the matricesA ∈ IRm×n and B ∈ IRk×n. The cells of these matrices are the coordinatesof points from A and B, respectively.

The problem of bilinear separation is to find two hyperplanesH1(ω

1, γ1) = x ∈ IRn | 〈ω1, x〉 = γ1, ω1 ∈ IRn, γ1 ∈ IR, andH2(ω

2, γ2) = x ∈ IRn | 〈ω2, x〉 = γ2, ω2 ∈ IRn, γ2 ∈ IR, separat-ing given sets A and B in the following sense.

D e f i n i t i o n 1. The sets A and B are said to be bilinearlyseparable by two hyperplanes H1(ω

1, γ1) and H2(ω2, γ2) if for each point

Ai (i = 1,m) of the set A, and for each point Bj (j = 1, k) of the set Bone of the following systems of inequalities holds:

〈Ai, ω1〉 > γ1,〈Ai, ω2〉 > γ2,

〈Bj , ω1〉 < γ1 or 〈Bj , ω2〉 < γ2;

(1)

〈Bj , ω1〉 > γ1,〈Bj , ω2〉 > γ2,

〈Ai, ω1〉 < γ1 or 〈Ai, ω2〉 < γ2;

(2)

〈Ai, ω1〉 > γ1,〈Ai, ω2〉 > γ2,

or

〈Ai, ω1〉 < γ1,〈Ai, ω2〉 < γ2,

〈Bj , ω1〉 > γ1,〈Bj , ω2〉 < γ2,

or

〈Bj , ω1〉 < γ1,〈Bj , ω2〉 > γ2.

(3)

It can be readily seen that the cases (1) and (2) are equivalent if theroles of A and B are interchanged.

199

The problem of bilinear separation in the sense of the system (1) canbe reduced to the following equivalent nonconvex bilinear optimizationproblem (non-symmetrical case) [2]:

F (z1, z2) = 〈z1, z2〉 ↓ minz1,ω1,γ1,z2,ω2,γ2

,

(z1, ω1, γ1) ∈ Z1 ,

(z1, ω1, γ1) ∈ R

K

∣∣∣∣∣∣

−Aω1 + γ1em + em ≤ 0,Bω1 − γ1ek + ek ≤ z1,

z1 ≥ 0,

(z2, ω2, γ2) ∈ Z2 ,

(z2, ω2, γ2) ∈ R

K

∣∣∣∣∣∣

−Aω2 + γ2em + em ≤ 0,Bω2 − γ2ek + ek ≤ z2,

z2 ≥ 0,

(BLP1)where K = k + n + 1, em , (1, 1, ..., 1) ∈ IRm, ek , (1, 1, ..., 1) ∈ IRk, z1and z2 are auxiliary variables (z1j = max

0, 〈ω1, Bj〉 − γ1

,

z2j = max0, 〈ω2, Bj〉 − γ2

, j = 1, k). These variables determine the

errors of separation. If z1 = 0 and z2 = 0 then sets A and B are non-symmetrically bilinearly separable. So, we have the following theorem.

Theorem 1. [2] The sets A and B are bilinearly separable in thespace IRn in the sense of the system (1) if and only if the value of theproblem (BLP1) is zero. In that case, the components (ω1

∗, ω2∗, γ

1∗ , γ

2∗) of

global minimum point determine separating hypeplanes 〈ω1, x〉 = γ1 and〈ω2, x〉 = γ2.

The problem (BLP1) is the bilinear optimization problem with disjointconstraints [3, 4].

In turn, the problem of bilinear separation in the sense of the system(3) can be reduced to the following optimization problem (symmetricalcase) [1]:

⟨(y1 + y2), (y3 + y4)

⟩+⟨(z1 + z2), (z3 + z4)

⟩↓ minyl,zl,ω1,ω2,γ1,γ2

,

−Aω1 + γ1em + em ≤ y1, −Aω2 + γ2em + em ≤ y2,Aω1 − γ1em + em ≤ y3, Aω2 − γ2em + em ≤ y4,Bω1 − γ1ek + ek ≤ z1, −Bω2 + γ2ek + ek ≤ z2,−Bω1 + γ1ek + ek ≤ z3, Bω2 − γ2ek + ek ≤ z4,

yl ≥ 0, zl ≥ 0, l = 1, 4,

(BLP3)where yl and zl (l = 1, 4) are auxiliary variables.

Similarly to Theorem 1, the sets A and B are symmetrically bilinearlyseparable if and only if the value of the problem (BLP3) is zero, and the

200

components (ω1∗, ω

2∗, γ

1∗ , γ

2∗) of global minimum point determine separating

hyperplanes.The problem (BLP3) is more difficult bilinear optimization problem

with joint constraints [4].3. Local and global search. It is clear that the nonconvexity of

the problems (BLP1) and (BLP3) occurs due to the scalar products ofindependent variables in the goal functions. In addition, it can be shownthat these functions can be represented as a difference of two convexfunctions, i.e. the goal functions are d.c. functions [5]. Therefore, for thedevelopment of methods for solving these problems one can applies theGlobal Search Theory in d.c. optimization [5].

Global Search Theory consist of two principal stages [5]:1) a special local search methods, which takes into account the structureof the problem under scrutiny;2) the procedures, based on Global Optimality Conditions [5], which allowto improve the point provided by the Local Search Method.

For example, consider the problem (BLP1). Since the objective func-tion F has a bilinear structure, so it is affine w.r.t. each of its variableswhen the other variable is fixed, for the implementation of local searchwe use well known idea of consecutive solving of the auxiliary linear pro-gramming problems, starting with fixed point (z1s , z

2s ) [3, 4]:

⟨z1, z2s

⟩↓ min

(z1,ω1,γ1),(z1, ω1, γ1

)∈ Z1,

⟨z1s , z

2⟩↓ min

(z2,ω2,γ2),(z2, ω2, γ2

)∈ Z2.

These auxiliary problems can be efficiently solved using known soft-ware packages. It is proved that Local Search procedure converges to thecritical point, which is partially global solution to the problem (BLP1)w.r.t. each group of variables

(z1, ω1, γ1

)and

(z2, ω2, γ2

). In practical

implementation, after a finite number of iterations when some stoppingcriterion is satisfied can be obtained a critical point with the requiredaccuracy.

Next, using the following d.c. decomposition:

F (z1, z2) = g(z1, z2)− f(z1, z2),

where f(z1, z2) =1

4

∥∥z1 − z2∥∥2 , g(z1, z2) =

1

4

∥∥z1 + z2∥∥2, and special

features of problem (BLP1) (see Theorem 1), an algorithm for global

201

search separating hyperplanes based on the method of solving bilinearprogramming of the general form [3, 4] was elaborated. It is necessaryto emphasize that for the construction of level surface approximation off(z1, z2), determining the basic nonconvexity in the problem (BLP1) (thisstage of the global search is one of the crucial stages), a new set of direc-tions, which are more effective than the standard sets [3, 4] is proposed.

As far as solving of the problem (BLP3) is concerned, we developthe same ideas of local and global search for the more difficult bilinearproblems with joint constraints.

4. Generation of test problems and computational simula-tion. To analyze the efficiency of new methods for solving optimizationproblems it is often required to be able to find or generate test problemswith known solutions and properties. Sometimes, libraries of test prob-lems are available, in some other cases, the problem under considerationhas a solution for any initial data (for example, this is the case of bimatrixgames [4]). In bilinear separation problems, one cannot guarantee thata solution exists for arbitrary initial data. Moreover, no representativetest libraries are available for these kind of problems. For that reason wepropose a new method for generating bilinear separation test problemswith known global solutions [6].

For example, to test the algorithms for local and global search in theproblem of (BLP1) more than 20,000 of various dimension (from 2 up to200) test problems were generated by proposed method. The programsthat implement the proposed algorithms of local and global search wereelaborated, and computational simulation was carried out. In total morethan 99% of the generated problems were solved with required accuracy.

In general, we can conclude that computational testing of the elabo-rated methods has shown the efficiency of the proposed approach to theproblems of bilinear separability.

References

1. O.L. Mangasaryan. “Linear and Nonlinear Separation of Patterns by LinearProgramming”, Operations Research, 13, No. 3, 444–452 (1965).

2. K.P. Bennet, O.L. Mangasaryan. “Bilinear Separation of Two Sets in n-Space”, Computational Optimization and Applications, 2, No. 3, 207–227(1993).

202

3. A.V. Orlov. “Numerical Solution of Bilinear Programming Problems”, Com-putational Mathematics and Mathematical Physics, 48, No. 2, 225–241(2008).

4. A.S. Strekalovsky, A.V. Orlov. Bimatrix Games and Bilinear Programming(in Russian), Fizmatlit, Moscow, Russia (2007).

5. A.S. Strekalovsky. Elements of Nonconvex Optimization (in Russian), Nauka,Novosibirsk, Russia (2003).

6. S.Yu. Pinigin. “The Method of Generation of Bilinear Separation Test Prob-lems”(in Russian), Abstracts of XII Baikal school-seminar of young researches“Modeling, optimization and information technologies”, 38 (2012).

Ensemble calculations application for estimation andoptimization of climate model parameters

Valeriy Parkhomenko

Institution of Russian Academy of SciencesDorodnicyn Computing Centre of RAS,

Moscow, Russia; [email protected]

A simplified climate model is presented which includes a fully three-dimensional, frictional geostrophic ocean component [1, 2] but retains anintegration efficiency considerably greater than extent climate models withthree-dimensional, primitive-equation ocean representations (3000 yearsof integration can be completed in about 5 hours on a PC). The modelalso includes energy and moisture balance atmosphere and a dynamic andthermodynamic sea-ice model.

Climate models incorporate a number of adjustable parameters whosevalues are not always well constrained by theoretical or observational stud-ies of the relevant processes. Even the nature of the processes may beunclear and dependent upon resolution, as sub-grid scale mixing param-eterizations, particularly for coarse resolution models, may represent awide variety of different physical processes (eddies and unresolved mo-tions, inertia-gravity waves, tides etc.). In such cases parameter valueswould ideally be chosen by optimizing the fit of model predictions to ob-servational data.

203

Efficient models have the potential to perform large numbers of inte-grations and hence explore larger regions of their parameter space. Wherethe parameters have clear physical interpretations or close equivalents inhigher resolution models, the results may be of more general relevance.Efficient models are also essential for the understanding of very long-termnatural climate variability, in which case the optimal inter-component bal-ance of model complexity may depend on the timescale range of interest.

Frictional geostrophic ocean models [1, 2] are applicable with arbitrarybottom topography in a global setting, but significantly simpler than theprimitive equation dynamics. We describe here the combination of thelatter model, an energy and moisture balance atmosphere and a dynamicand thermodynamic sea-ice model. At a resolution of 72 by 72 cells inthe horizontal [3], and given the extremely simple representation of theatmosphere, the resulting coupled model is highly efficient. We perform aninitial investigation of the space generated by the simultaneous variationof model parameters by analyzing a set of model runs. We do not attemptto produce well-converged statistical analyses, our aim is to investigate theextent to which both model parameters and model predictions of globalchange are constrained by quantitative comparison with data [4]. Thuswe commence our analysis by defining and applying an objective measureof model error and discuss the modeled climate in the low-error runs.

The frictional geostrophic ocean model principal governing equations[1, 2] are similar to classical general circulation models, with the neglect ofmomentum advection and acceleration. In the vertical there are normally8 depth levels on a uniformly logarithmically stretched grid with verticalspacing increasing with depth from 175 m to 1420 m. The maximumdepth is set to 5 km. The horizontal grid is uniform in the longitude andsin of latitude coordinates giving boxes of equal area in physical space.The horizontal resolution is normally 72 by 72 cells [3].

We use an energy and moisture balance model of the atmosphere,similar to that described in [5]. The prognostic variables are surface airtemperature and surface specific humidity for which the governing equa-tions can be written. The short-wave solar radiative forcing representsseasonally changed conditions. In a further departure from that model,the relevant planetary albedo is given by a surface type properties. Oversea ice the albedo is temperature-dependent. Heat absorption by watervapor, dust, ozone, clouds, etc. is parameterized by the constant value.The remaining heat sources and sinks describe the long-wave radiation

204

and latent heat processes. For anthropogenically forced experiments agreenhouse warming term is added which is proportional to the log of therelative increase in carbon dioxide concentration as compared to an arbi-trary reference value. The sensible heat flux depends on the air-surfacetemperature difference and the surface wind speed (derived from the oceanwind-stress data), and the latent heat release is proportional to the pre-cipitation rate [5].

It is used an implicit scheme to integrate the atmospheric dynami-cal equations. The scheme comprises an iterative, semi-implicit predictorstep followed by a corrector step which renders the scheme exactly con-servative. Sea ice dynamical equations are solved for the fraction of theocean surface covered by sea ice in any given region and for the averageheight of sea ice. In addition a diagnostic equation is solved for the surfacetemperature of the ice. Following [6, 7] thermodynamic growth or decayof sea ice in the model depends on the net heat flux into the ice from theocean and atmosphere. Sea-ice dynamics simply consist of advection bysurface currents and Laplacian diffusion with constant coefficient.

Fixing the distribution of drag, we have a set of 10 model parametersrelated to mixing and transport, augmented to 12 if we allow for variationof the width and slope and of the atmospheric diffusivity. If we vary theseparameters individually, as in conventional, single parameter studies, wevisit only a very restricted region of parameter space. We therefore allowall 12 parameters to vary at once within specified ranges [4].

Note that it may be appropriate to use larger values of frictional anddiffusive parameters than in higher resolution models. In our semi-randomapproach, we generate an ensemble by uniformly spanning the range ofeach individual parameter, but choose combinations of parameters at ran-dom. This is equivalent to an equal subdivision of probability space if theprobability distributions for the parameters are uniform. Thus with Mruns and N parameters, each parameter takes M, uniformly (or logarith-mically) spaced values between its two extrema, but the order in whichthese values are taken is defined by a random permutation. Each runis a separate, 2000-year integration from a uniform state of rest understandart forcing. We are primarily concerned with the effects of vari-ations of model parameters, thus we fix the initial ocean temperatureat 20 C unless otherwise stated. This results in a rapid, convectivelydriven start to the oceanic adjustment process. To process the results ofa large number of runs we have to define an objective measure of model

205

error. To do so it is used a weighted root mean square error over theset of all dynamical variables in the ocean and atmosphere, as comparedto interpolated observational datasets, namely NCEP surface (1000 mb)atmospheric temperature and specific humidity, averaged over the period1948 to 2002, and ocean temperature and salinity [8].

By analyzing a randomly generated set of runs, each 2000 years inlength, we have considered the uncertainty in 12 mixing and transportparameters. Constructing a quantitative measure for the model errorallowed us to address both the inverse problem of estimation of model pa-rameters, and the direct problem of model predictions. Results representan attempt at tuning a 3-D climate model by a strictly defined proce-dure which nevertheless considers the whole of the appropriate parameterspace. Modelling approach is thus to match model outputs to observationswhile model inputs (parameters) are initially only weakly constrained.


(projects no. 11-01-93003, 11-01-00575, 11-07-00161) and Presidium RAS Basic

Research Program (no. 14).

References

1. N.R. Edwards, J.G. Shepherd. “Bifurcations of the thermohaline circulationin a simplified three-dimensional model of the world ocean and the effects ofinterbasin connectivity,” Clim Dyn, No. 19, 31-42 (2002).

2. V.P. Parkhomenko. “Climate model with consideration World ocean deepcirculation,” Vestnik MGTU im. Baumana, Issue Mathematical Modelling,p. 186-200 (2011) (in Russian).

3. V.P. Parkhomenko, I.S. Kachurina. Numerical experiments with coupledAGCM and thermohaline ocean circulation model, Applied mathematics Re-ports, CC RAS, Moscow, (2011) (in Russian).

4. N.R. Edwards, R. Marsh. “Uncertainties due to transport-parameter sensi-tivity in an efficient 3-D ocean-climate model,” Climate Dynamics, No. 24,415-433 (2005).

5. A.J. Weaver, M. Eby, E.C. Wiebe, C.M. Bitz, P.B. Duffy, T.L. Ewen, A.F.Fanning, M.M. Holland, A. MacFadyen, H.D. Matthews, K.J. Meissner, O.Saenko, A. Schmittner, H. Wang, M. Yoshimori. “The UVic Earth SystemClimate Model: Model description, climatology, and applications to past,present and future climates,” Atmos-Ocean, No. 39, 361-428 (2001).

6. W.D. Hibler. “Dynamic thermodynamic sea ice model,” J Phys Oceanogr,No. 9, 815-846 (1979).

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7. A.J. Semtner. “Model for thermodynamic growth of sea ice in numericalinvestigations of climate,” J Phys Oceanogr, No. 6, 379-389 (1976).

8. S. Levitus, T.P. Boyer, M.E. Conkright, T. OBrien, J. Antonov, C. Stephens,L. Stathoplos, D. Johnson, R. Gelfeld. Noaa Atlas Nesdis 18, World oceandatabase 1998, vol. 1, Introduction, US Government Printing WashingtonDC, (1998).

Interior point algorithms

Sergey Perzhabinsky1, Valery Zorkaltsev2

1 Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia;

[email protected] Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia;

[email protected]

Introduction. The algorithms for mathematical programming prob-lems are considered in the report. Solution improvement in these algo-rithms is carried out inside set of strongly feasible points. The stronglyfeasible points are points which satisfy inequality constraints in strictform. The pioneer results of the algorithm studies were obtained by S.Antsyz, I. Dikin [1], Yu. Evtushenko [2], V. Zhadan, V. Zorkaltsev. Thegiven interior point algorithms are effectively used for realization of anapplied models since 70th years of the last century. The interesting prop-erties of algorithms were discovered on the basis of the theoretical andexperimental studies of the algorithms. In particular, it was shown thatthe given algorithms are produced relative interior point of the optimalsolution set in case of non-uniqueness of the optimal solution. The rela-tive interior point has minimal set of the active constraints. The interiorpoint algorithms attract the heightened interest since the middle of 80thyears of the last century. It was demonstrated in the independent studiesthat the interior point methods are more efficient for linear programmingproblems comparing with the simplex-method. In addition the algorithmsfor solving nonlinear programming problems can be easy realized on thebasis of the interior point method.

The results of theoretical justification and experiments of the interiorpoint algorithms for solving linear are presented in the report. The ways

207

of improving of computational efficiency of the interior point algorithmsare considered. The results of some studies of the interior point algorithmsfor nonlinear programming are discussed.

Let’s consider the primal and dual linear programming problems

cTx→ minx∈X

, X = x ∈ Rn : Ax = b, x ≥ 0 ; (1)

bTu→ maxu∈U

, U = u ∈ Rm : g (u) ≥ 0 . (2)

Here g (u) = c − ATu is linear vector-function with components gj (u),j = 1, . . . , n. The vectors x ∈ Rn, u ∈ Rm are variables of problems(1), (2). The vectors c ∈ Rn, b ∈ Rm and the matrix A of order n ×mare given.

The vectors x ∈ X , u ∈ U are feasible solutions of problems (1), (2).The sets of optimal solutions of these problems are denoted

X = ArgmincTx : x ∈ X

, U = Argmax

bTu : u ∈ U

.

The problem (1) is the primal linear programming problem. The problem(2) is the dual linear programming problem.

The problem (1) is called non-degenerate, if there is no more one vectoru ∈ Rm for which complementary conditions hold for any x ∈ X :

xjgj (u) = 0, j = 1, . . . , n. (3)

The problem (2) is called non-degenerate, if there is the unique x ∈ Rnsuch that Ax = b and condition (3) holds for any u ∈ U .

The interior region of the convex set Q ⊂ Rn relative to minimal linearmanifold which contains this region, is called relative interior riQ [3].

The class of interior point algorithms. The initial approximationx0 > 0 is given. The iterative process is considered

xk+1 = xk + λk∆xk, k = 0, 1, 2, . . . . (4)

Here the vector ∆xk is the direction of improving the solution on iterationk, the value λk is the positive step in this direction. The vector ∆xk isthe solution of the auxiliary problem

cT∆x+1

2

n∑

j=1

(∆xj)2

dkj→ min

∆x∈Rn, (5)

208

A∆x = rk, (6)

whererk = b−Axk. (7)

In addition the values 1/dkj should be interpreted as penalty coefficientswhich prevent breaking the borders of the feasible solution set. The inversevalue of penalty coefficient will be called weighting coefficient.

The interval and axiomatic approaches to defining of coefficients dkjwere introduced in [4]. The inequalities must be hold for the weightingcoefficient

σj(xkj)≤ dkj ≤ σj

(xkj), j = 1, . . . , n (8)

where σ, σ are some continuous non-decreasing functions of positive ar-gument which satisfies the two conditions

0 < σ (t) ≤ σ (t) , ∀t > 0, (9)

σ (t) ≤Mt (10)

for some ε > 0, M > 0 and all t ∈ (0, ε]. The rules of definition of valuesdkj which satisfy (8) - (10) forms the class of algorithms. In particular,the weighting coefficients can be given in the form of functions of thecomponents of vector of variables

dkj =(xkj)p, j = 1, . . . , n (11)

where p ≥ 1. Let’s denote Dk = diag(dk)where dk is the vector of Rn

with components dkj , j = 1, . . . , n.The step size is computed by the formula

λk = γk minj: ∆xk

j<0

−xkj∆xkj

(12)

as γk ∈ (0, 1). The meanings of parameter γk doesn’t decrease on iter-ations γk ≤ γk+1 for all k. The moving from point xk by the direction∆xk is realized on the part of way which is equal γk to the bound of theregion x ≥ 0. It is possible to change the parameter γk at each iteration.In particular, the parameter γk can converge to 1. This rule can be usedfor the acceleration of the computational process.

209

Using the Lagrange method of multipliers for solving the problem (5),(6), we obtain

∆xk = −Dkg(uk)

where the vector uk is the solution of the system of linear equations

ADkATu = rk +ADkc. (13)

Thus the vector of the Lagrange multipliers of the auxiliary problem isfound at each iteration. This vector is an approximation of the solutionof the dual linear programming problem.

The iterative process of the improving of the solution of problem (1)consists of the two stages. The first stage is the input to the feasibleregion. The absolute values of the residuals of the equality constraintsmonotonically decrease on the iterations at this stage. It follows from (4),(6), (7) that rk+1 = (1− λk) rk. Therefore if rk 6= 0 then

λk = min1, λk (14)

where λk is computed by (12).The optimization in the feasible region is the second stage of the it-

erative process. All constraints of the problem (1) are satisfied and theobjective function value monotonically decreases in this case. In additionrk is equal 0 in (13).

The theoretical justification of the given computing process in thefeasible region is presented in [5]. This theorem is proved under the non-degeneracy condition and for the different weighting coefficients.

Theorem 1. Let the problem (1) is non-degenerate, X 6= ∅, X 6= Xand x0 ∈ X then the following statements are true for the given algorithm.1. There are x ∈ X, u ∈ riU such that

xk → x, uk → u as k →∞.

2. If the condition holds for the weighting coefficients dkj

σ (α) /σ (β) = O (α/β) (15)

then x ∈ riX , the values∥∥uk − u

∥∥,∥∥xk − x

∥∥ converge to 0 no slower thanarcwise, and the following relation takes place for some positive values P1,P2, P3, P4, ρ ∈ (0, 1) for all k∥∥uk − u

∥∥ ≤ P1Lk, Lk ≤ P2Tk, Tk ≤∥∥xk − x

∥∥ ≤ P3Tk, Tk ≤ P4 (ρ)k

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whereLk = max

j: gj(u) 6=0dkj , Tk = max

j: gj(u)6=0xkj .

3. If the following condition holds for the weighting coefficients dkj

σ (α) /σ (β) = o (α/β) (16)

then ∥∥uk − u∥∥ /∥∥xk − x

∥∥→ 0 as k →∞ (17)

for some P5 > 0 and all k

Tk+q+1 ≤ P5Tkmax(1− γτ ) : τ ∈ k, . . . , k + q.

Here q is amount of numbers j such that gj (u) 6= 0.Let’s note that the condition (15) holds for the weighting coefficients

(11) as p = 1. If p > 1 then the stronger condition (16) holds. Thealgorithm with the weighting coefficients (11) as p = 2 is also known in thewestern publications as affine scaling method. The weighting coefficients

dkj =(xkj)2

and the step λk =((g(uk))T

D−1k g

(uk))1/2

are used in Dikin

interior point algorithm. If X 6= ∅ then vectors xk converge arcwise to apoint of riX in this algorithm [6].

The theoretical justification of the algorithms without the non-degenerateassumption is very important for the theory and practice. The given algo-rithms with the weighting coefficients (11) for all p ∈ (1, 3] were theoreti-cally justified without the non-degenerate assumption [7]. The additionalcondition on γk was used for this justification

γk ∈ (0, 2/(p+ 1)), k = 0, 1, 2, . . . . (18)

Theorem 2. Let X 6= ∅, X 6= X and x0 ∈ X then the followingstatements are true for the algorithm (4) - (12) as p ∈ (1, 3] and thecondition (18).1. If X 6= ∅ then there are x ∈ riX, u ∈ riU such that

xk → x, uk → u,

∥∥xk+1 − x∥∥ /∥∥xk − x

∥∥→ (1 − γ),∥∥uk − u

∥∥ /∥∥xk − x

∥∥→ 0 as k →∞.

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2. If X = ∅ then limk→∞

cTxk = −∞ or the all components of the vector

∆xk are positive of some iteration k.Conclusion. The condition (8) allows us to use different techniques

of computing of the weighting coefficients. It is not necessary to havespecific expressions for the functions σ, σ. It is enough to have the proofof the existence of such functions and execution of their properties. Inparticular, the following weighting coefficients are effective under k ≥ 1

dkj =xkj

maxε, gk−1j

, j = 1, . . . , n (19)

where ε is small positive value. The interior point algorithms with theweighting coefficients (19) have superlinear rate of convergence [5].

The execution of the condition (10) for the weighting coefficients isenough for convergence of computational process accordingly to Theorem1. The execution of the condition (15) is necessary for superlinear conver-gence of computational process and the receiving of the relative interiorpoint of the optimal solution set. The execution of the condition (16) andconvergence γk to one as k → ∞ provide superlinear rate of convergenceof the given algorithm.

If the relation (17) holds then vectors of dual variables converge tothe optimal solution of the dual problem rather than the primal variablesconverge to their optimal meanings.

References

1. I.I. Dikin. “Iterative solution of problems of linear and quadratic program-ming”, Sov. Math. Dokl., Vol. 8, 674–675 (1967) (in russian).

2. Yu.G. Evtushenko, V.G. Zhadan. “Numerical methods for solving some op-erations research problems”, Comput. Maths. Math. Phys., Vol. 13, No. 3,56–77 (1973) (in russian).

3. R.T. Rockafellar. Convex analysis, Princeton University Press, Prinston(1970).

4. V.I. Zorkaltsev. Relative interior point method, Komi Branch of Academy ofSciences of the USSR, Syktyvkar (1986) (in russian).

5. V.I. Zorkaltsev. “On a class of interior point algorithms,” Comput. Math.Math. Phys., Vol. 49, No. 12, 2017–2033 (2009).

6. I.I Dikin. “On a convergence of the iterative process,” Controlled Systems,No. 12, 54–60 (1974) (in russian).

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7. V.I. Zorkaltsev. The justification of the class of projective algorithms, Melen-tiev Energy Systems Institute SB RAS, Irkutsk (1995) (in russian).

Interactive optimization as a tool for finding thecomplex periodic solutions in nonlinear dynamics

Lev F. Petrov1

1 Plekhanov Russian University of Economics, Department of Economics and

Mathematics, Moscow, Russia; [email protected], [email protected]

I n t r o d u c t i o n. We consider essentially nonlinear dynamicalsystems with the ability to implement a chaotic behavior and deterministicsolutions of various kinds. Among the deterministic solutions, we willhighlight a variety of periodic solutions of different periods. Problems ofcontrol of dynamic regimes in such systems discussed in [1]. This workis devoted to numerical algorithms for constructing and analyzing thestability of periodic solutions of strongly nonlinear dynamical systems.

P r o b l e m S t a t e m e n t. We will consider the strongly non-linear system, which places no restrictions on the value of the individualcomponents. In the framework of this approach we can analyze the lin-ear and quasi-linear system, but the focus will be given to an essentiallynonlinear systems of general form. The only requirement that we maketo a dynamic system is the ability to construct numerical solutions of theCauchy problem with the required precision.

We will use a general approach to the problem of constructing periodicsolutions of nonlinear systems of ordinary differential equations. Thisapproach H. Poincare [2] formulated as follows: Suppose that

dxi(t)/dt = Xi (i = 1, 2, ..., n) (1)

is the system of differential equations, where Xi - data, unambiguousfunction of the variables x1, x2, ..., xn, and maybe, time t.

Suppose now that

x1 = ϕ1(t), x2 = ϕ2(t), ..., xn = ϕn(t) (2)

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is particular solution of this system.Imagine that at time T n variables xi take their initial values, so that

ϕi(0) = ϕi(T ). It is clear that at time T , we will be in the same conditionsas at time 0, and hence for any t ϕi(t) = ϕi(t + T ). In other words, thefunctions ϕi(t) are periodic functions of t.

Variant 1. The system (1) is autonomous, that is, the right parts Xi

are not depend on time t. In this case, the period T of the solution isunknown.

Variant 2. The system (1) is non-autonomous, that is, the right partsXi are depend on time t:

Xi = Xi(t, x1, x2, ..., xn) (3)

In this case, the period T of the system (1) is known:

Xi(t, x1, x2, ..., xn) = Xi(t+ T, x1, x2, ..., xn) (4)

A periodic solution can have a multiple of the period kT, k = 1, 2, :

ϕi(t) = ϕi(t+ kT ) (5)

H. Poincare for finding periodic solutions of the system implies theexistence of a small parameter. We use his approach on the initial con-ditions of the periodic solution, but let’s not assume the existence of asmall parameter in the system (1).

Problem Statement of constructing periodic solutions of strongly non-linear autonomous dynamical system (1) (variant 1):

Find the initial conditions ϕi(0), (i = 1, ..., n), corresponding to theperiodic solution and the period T of this solution: ϕi(0) = ϕi(T ), (i =1, ..., n), and therefore ϕi(t) = ϕi(t+ T ), (i = 1, ..., n).

Problem Statement of constructing periodic solutions of strongly non-linear autonomous dynamical system (1) (variant 2):

Find the initial conditions ϕi(0), (i = 1, ..., n), corresponding to thekT -periodic solution: ϕi(0) = ϕi(kT ), (i = 1, ..., n, k = 1, 2, ...), andtherefore ϕi(t) = ϕi(t+ kT ), (i = 1, ..., n).

Note that the dimension of variant 1 is n+1, the dimension of variant2 is n. After finding the initial conditions of the periodic solution it isbuilt using the numerical integration in one period.

Consider the algorithm for determining the initial conditions of theperiodic solution of nonautonomous nonlinear dynamics problems (variant

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2). The period T of the system is known. The period of solutions givenby integer parameter k at the start. Note that the increase in the integerparameter k does not significantly restrict the form of the solution. Forexample, for k = 12 in the search box, the solution with k = 1, 2, 3, 4, 6, 12are included. (Fig.1).

Fig. 1

Denote the unknown initial conditions, corresponding to kT -periodicsolution Yi = xi(0). Obviously, for periodic solutions to satisfy the equal-ity

xi(kT ) = Yi, i = 1, 2, ..., n (6)

This is done only when the initial conditions corresponding to a peri-odic solution.

Consider the function (Fig.2)

F (Y1, Y2, ..., Yn) =

√√√√n∑

i=1

[Yi − xi(kT )]2 (7)

Fig. 2

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This function defines the discrepancy in fulfilling the conditions ofperiodicity. Obviously, for a periodic solution

F (Y1, Y2, ..., Yn) = 0 (8)

Therefore, to determine the initial conditions corresponding to a pe-riodic solution, it is possible to use optimization algorithms with the ob-jective function

F (Y1, Y2, ..., Yn)− > min (9)

Comparative numerical experiments have shown that the more effec-tive is another way to determine the initial conditions Yi, i = 1, 2, ..., ncorresponding to a periodic solution.

To find the initial conditions Y1, Y2, ..., Yn corresponding a periodicsolution we use a system of nonlinear algebraic equations

Yi − xi(kT ) = 0, i = 1, 2, ..., n (10)

This system is not divided into separate equations, since the quantitiesxi(kT ) are determined from the original nonlinear system of ordinary dif-ferential equations (1) by numerical calculation of the Cauchy problem onthe interval [0, kT ]. To solve this system we have used Newton’s method.In the computer implementation of this algorithm includes the possibilityof interactive control of calculations. This allows us to specifically controlthe calculations to find the most interesting of periodic solutions.

For an autonomous system of ordinary nonlinear differential equations(1) (variant 1) the period T of the solution is also unknown. In this case,since the initial time is arbitrary, we assume that Yn = 0. Then to findthe initial conditions of Y1, Y2, ..., Yn−1 and period T of solutions we havethe system of nonlinear algebraic equations

Yi − xi(T ) = 0, i = 1, 2, ..., n− 1 (11)

xn(T ) = 0

This system is solved using Newton’s method. After finding the ini-tial conditions Y1, Y2, ..., Yn−1 and period T of solution we numericallycalculate periodic solution itself.

Note that the dimensions of systems of nonlinear algebraic equationsfor variant 1 (11) and variant 2 (10) are the same and equal to n.

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To determine the stability of periodic solutions found, we constructthe variational system and calculate the multipliers.

Note that the presented algorithm for finding periodic solutions ofstrongly nonlinear dynamical systems is iterative, and stability analysisalgorithm is finite.

I n t e r a c t i v e a l g o r i t h m. The complexity of the dynamicsystem behavior is described by periodic solutions of strongly nonlinearsystems of ordinary differential equations determines the branching struc-ture of the algorithm for constructing periodic solutions. Here we inves-tigate the evolution of these solutions when changing parameters of thedynamical system.

Note that the convergence of Newton’s method depend on startinginitial conditions. These conditions can be defined either randomly or onthe results of the previous step on the parameter of the dynamical system.The dynamical system may not have periodic solutions of a certain pe-riod. To overcome these and other computational problems, our algorithmfor finding periodic solutions of nonlinear systems of ordinary differentialequations is realized in an interactive mode (Fig. 3). Interim results offinding periodic solutions are shown to the user in real time. The usercan intervene in the computation and change the parameters of dynamicsystems, finding solution numerical methods, tactics of searching.

This approach allows us to find most complex periodic solutions ofnonlinear dynamical systems with several degrees of freedom even in thefield of dynamical chaos, where all the periodic solutions are unstable.Also, this approach allows us to investigate bifurcation of periodic solu-tions.

Fig. 3

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References

1. L.F. Petrov. “Control of dynamical regimes of systems with deterministicchaos” Abstracts of reports at II International conference Optimization andapplications (Optima-2011), 175 – 178 (2011).

2. Henri Poincare. Selected Works in 3 volumes. (Russian). V.1. New methodsof celestial mechanics, Nauka, Moscow, 1971, 772 p.

3. L.F. Petrov. “Nonlinear effects in economic dynamic models,” NonlinearAnalysis, 71 , 2366–2371 (2009).

On the Semidefinite Representation of the MaximumOptimal Rate Problem in LDPC Codes

M.R. Peyghami1, H. Tavakoli2, M. Ahmadian Attari3

1 Department of Mathematics, K.N. Toosi University of Technology, P.O. Box

16315-1618, Tehran, Iran; [email protected] Department of Electrical Engineering, K.N. Toosi University of Technology,

Tehran, Iran; m [email protected] Department of Electrical Engineering, K.N. Toosi University of Technology,

Tehran, Iran; [email protected]

Abstract

Achieving and approaching to the channel capacity are two ma-jor important properties of designing good LDPC codes. For de-signing good LDPC codes in infinite mode, it is important to solvethe maximum optimal rate problem. The Binary Erasure Channel(BEC) consists of the simplest form of the optimal rate problem,and achieving and approaching to the capacity for BEC have beenstudied in literature by many researchers. Although, the structureof the optimal rate problem for BEC is the simplest form, but find-ing an answer to satisfy the non-linear constraints of the problem isa hard part of the problem. In this talk, we will discuss about thesemidefinite reformulation of the optimal rate code design in BEC.

Keywords: Semidefinite Optimization, Binary Erasure Channel, Max-imum Optimal Rate Problem.

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Problems of optimal resistance in Newtonianaerodynamics

Alexander Plakhov1

1 University of Aveiro, Aveiro, Portugal and Institute for Information

Transmission Problems, Moscow, Russia; [email protected]

A body moves in a rarefied medium of resting particles and at the sametime very slowly rotates (somersaults). Each particle of the medium isreflected elastically when hitting the body boundary (multiple reflectionsare possible). The resulting resistance force acting on the body dependson the time; we are interested in minimizing the time-averaged value ofresistance (which is called R). The value R(B) is well defined in terms ofbilliard in the complement of B, for any bounded body B ⊂ R

d, d ≥ 2with piecewise smooth boundary.

Let C ⊂ Rd be a bounded convex body and C1 ⊂ C be another convex

body with ∂C1 ∩ ∂C = ∅. It would be interesting to get an estimate for

R(C1, C) = infC1⊂B⊂C

R(B). (1)

If ∂C1 is close to ∂C, problem (1) can be referred to as minimizing theresistance of the convex body C by ”roughening” its surface. We cannotsolve problem (1); however we can find the limit

limdist(∂C1,∂C)→0

R(C1, C)

R(C). (2)

It will be explained that problem (2) can be solved by reductionto a special problem of optimal mass transportation, where the initialand final measurable spaces are complementary hemispheres, X = x =(x1, . . . , xd) ∈ Sd−1 : x1 ≥ 0 and Y = x ∈ Sd−1 : x1 ≤ 0. Thetransportation cost is the squared distance, c(x, y) = 1

2 |x − y|2, and themeasures in X and Y are obtained from the (d−1)-dimensional Lebesguemeasure on the equatorial circle x = (x1, . . . , xd) : |x| ≤ 1, x1 = 0 byparallel translation along the vector e1 = (1, 0, . . . , 0). Let C(ν) be thetotal cost corresponding to the transport plan ν and let ν0 be the trans-

port plan generated by parallel translation along e1; then the value inf C(ν)C(ν0)

coincides with the limit in (2).

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Surprisingly, this limit does not depend on the body C and dependsonly on the dimension d. In particular, if d = 3 (d = 2), it equals(approximately) 0.96945 (0.98782). In other words, the resistance of a3-dimensional (2-dimensional) convex body can be decreased by 3.05%(correspondingly, 1.22%) at most by roughening its surface.

The author was supported by FEDER funds throughCOMPETE–Operational

Programme Factors of Competitiveness and by Portuguese funds through the

Center for Research and Development in Mathematics and Applications (CIDMA)

and the Portuguese Foundation for Science and Technology (FCT), within

project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-

FEDER-022690; by the FCT research project PTDC/MAT/113470/2009; and

by the Grant of President of Russia for Leading Scientific Schools NSh-5998.2012.1.

L1 problems in control and numerical methods fortheir solution

Boris Polyak1

1 V.A. Trapeznikov Institute of Control Sciences RAS, Moscow, Russia;

[email protected]

L1 techniques is highly popular in many fields, it suffices to mentionLasso regression, exact penalties and basis pursuit in optimization, Leastabsolute values methods in estimation, Compressed Sensing in signal andimage processing, SVM methods in classification and recognition. How-ever this approach was not exploited in control theory.

We consider various problems in linear control theory which can betreated in the framework of L1-optimization. They include optimal con-trol with L1-performance index, L1-filtering, feedback stabilization withreduced number of outputs or states available. For them we study theproperties of the solutions and provide appropriate numerical methods.

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Object-oriented Framework for Dynamic Control ofthe Parallel Branch-and-Bound

Mikhail Posypkin1, Izrael Sigal2

1 Institute for Systems Analysis of RAS, Moscow, Russia;

[email protected] Dorodnicyn Computing Centre of RAS, Moscow, Russia; [email protected]

The Branch-and-Bound (B&B) is a general name for a family of meth-ods to split an initial problem into subproblems which are sooner or latereliminated by bounding rules. Bounding rules determine whether a sub-problem can yield a solution better than the best solution found so far.Today there is a big variety of Branch-and-Bound methods applicable tosolving discrete, mixed-integer and continuous global optimization prob-lems and even to multiobjective problems[1,2].

B&B methods for different problems share the common structure.Thus it is possible to select the set of parameters driving the resolu-tion process common for different problems. Below we outline the mostimportant parameters:

• the subproblem selection strategy (width-first, best-first, depth-firstor a combined search strategy);

• heuristic to improve the incumbent solution (0 for none, or the pos-itive number for selecting the respective heuristic);

• the bounding strategy (0 for none and positive number for selectingthe respective bounding strategy).

In [1] it was shown that the proper selection of the mentioned parame-ters can speed up the resolution process or improve the obtained solution.It is possible to select these parameters statically or adjust them dynam-ically during computations. In the latter case we can express the controlof computations as a extended finite state machine (EFSM) that acceptsthe characteristics of the computation process as an input and issues com-mands driving the computation process, e.g. ”set depth-first strategy”.

In the parallel implementation parallel processes exchange the sub-problems and incumbents with each other. So besides the commands

221

applicable to the sequential case we have some more commands specificto the parallel implementation:

• send N subproblems to the process P;

• send incumbent to the process P;

• send control command to the process P;

• recieve information (subproblems, incumbent or control command)from the process P.

The parallel B&B method’s logic can be defined as a EFSM that issuesone the commands described above as a reaction on certain events. Theexamples of events are:

• the requested number of B&B steps performed;

• the incumbent was updated;

• data arrived from some process.

It is worth noting that this set of commands is problem-independent.And thus it is possible to separate the logic of the B&B method defined bythe EFSM and the problem specific implementation of those commands.Such separation is important for several reasons. First, it saves effortswhen implementing new problem because only the problem-specific parthas to be implemented and the B&B logic is reused. Second, generalpart strictly defined using EFSM language can be a subject for a separatestudy. For instance it is possible to compare different load balancingstrategies on a simulator or check the correctness of the parallel algorithm,e.g. identify possible deadlocks.

The proposed approach was implemented in BNB-Solver library[3].BNB-Solver is written in C++. The class hierarchy can be roughly splitinto several different groups:

• basic numerical subroutines;

• communication subroutines supporting data exchange over MPI;

• problem specific modules;

• computations control modules (schedulers);

222

• solver classes.

The concrete application for solving certain optimization problem isobtained by combining three classes: the scheduler class, the problem-specific resolver class and the communicator class. The scheduler classmanages the computations by issuing commands to either resolver or com-municator. The resolver performs the requested number the computationsteps. The communicator class carries out data exchange between dif-ferent processes. Currently this approach was tested on the knapsackproblem, mixed-integer programming problems and multiobjective opti-mization.

This work was supported by Russian Foundation for Basic Research (project

no. 11-07-00360).

References

1. I.Kh. Sigal, A.P. Ivanova Introduction to Applied Discrete Optimization:Models and Computational Algorithms, Nauka, Moscow (2007).

2. Yu. Evtushenko, M. Potapov. “Methods of numerical solutions of multicrite-rion problems.,” Soviet Maths., Doklady, Vol. 34, pp. 420-423 (1985).

3. Y. Evtushenko, M. Posypkin, I. Sigal. “A framework for parallel large-scaleglobal optimization,” Computer Science - Research and Development Vol. 23No. 3, pp. 211-215, (2009).

Graf of decision logistics making for problem of goodsdelivery

Ekaterina Rassadnikova1, Aida Valeeva2, NelliMagafurzyanova3

1 Ufa State Aviation Technical University,Ufa, Russia;

[email protected] Ufa State Aviation Technical University,Ufa, Russia;

[email protected] Ufa State Aviation Technical University,Ufa, Russia;

[email protected]

The largest industry of our country is oil refining industry. Currentlyin Russia There are 30 large oil refineries with a total production capacity

223

of 261.6 million tons, and 80 mini-refineries with total refining capacity of11.3 million tons. What are costs faced by the oil refinery? The cost ofpreparation of oil products connected with the specific conditions of pro-duction, training, transportation and refining of oil in the country. Thatis to say to costs of oil products preparation was minimal, it is necessarynot only to optimize the processes associated with the manufacture ofproducts in a particular enterprise, but also the processes occurring inthe accompanying enterprises. Namely optimization of an accompanyingenterprise is considered in this article.

A company produces two types of a reagent used to neutralizationof hydrogen sulphide and mercaptan in the gaseous and liquid mediums.Two types of raw materials are required for the production of the reagent,which the company purchases in several Russian cities. The reagent isproduced on the installation a maximum production power of 25 tons perday. Consumers of the reagent are refineries in Russia. Warehouse is usedfor the storage of raw materials and finished products, this warehouselocated on the territory of the enterprise (storage capacity is 200 tons).

The company is interested in the organization of an effective managethe entire supply chain of the reagent, which includes categories: pro-curement and supply management, inventory management, production,warehousing, product packing, material handling, order processing.

All categories presented above are related. In order to obtain an ef-fective supply chain management these categories represented in the formof a graph (Figure 1). This graph was compiled on the basis of the graphlogistics solutions presented in [1].

Currently our group has considered following categories of decision-making logistics Demand Forecasting (vertex 8), Product packing (vertex32) and Transportation (vertices 23-30).

Category Transportation consists of the following tasks:23. Transportation modes24. Types of carriers25. Carriers26. Degree of consolidation27. Transportation fleet mix28. Assignment of customers to vehicles29. Vehicle routing and scheduling

224

30. Vehicle load plans

Fig. 1 - Graf logistics solutions for the enterprise that produces andsupplies the reagent. Vertices - logistical problems and directed edges -

connections between them.

Let us consider how these categories are related. We can make furtherdecisions about the desired level of reserves (12) and a transportation fleetmix (27) based on the forecasting demand,. Adopted decisions in its turninfluence on the subsequent decisions, directly or indirectly. Thus, thedesired inventory level directly affects the reserve stock (13). And decisionabout inventory control strategy (9) affects the relative importance ofstocks (10) indirectly, through a decision about the supplier (20). Relativeimportance of stocks (10) impacts on management methods (11). Thus inaddition to decisions within the logistics system, we consider some factorsthat ”come” from outside.

Influence of tasks on the transportation problems category (vertices23-30) can be seen on Figure 2.

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Fig. 2 - Influence of tasks on the transportation problems category(vertices 23-30).

The selected package (vertex 32) influences warehouse layout (vertex41), order entry procedures (vertex 45) and material handling (vertex 39).

Consider a example delivery of petrochemical product (reagent). It isnecessary to deliver the reagent produced in Ufa to Astrakhan. Thus, theproblem of delivery to consumers has the following optimality criteria:

1) Demand forecasting2) Choice of way and means of transportation3) Optimal route4) Choice of container type5) Exploitation of the loading space.You can see solution of this problem on Figure 3.

This article was examine the problems associated with the delivery ofthe goods in the containers by road routes in different cities of Russia.We have presented a problem in the form of a graph making logisticaldecisions.


(project no. 11-07-00579, 12-07-00631).

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Fig. 3 - Solution of real problem

References

1. Andr Langevin, Diane Riopel Logistics systems: design and optimization,Springer(2005).

2. N.I. Yusupova, A.F. Valeeva, E.Y. Rassadnikova, I.M. Latypov, I.S.Koshcheev “Multicriteria problem of cargo delivery to different consumersl,”Logistics and Supply Chain Management, 1, No. 5, 60-82 (2011).

Discontinuous control and Lyapunov functions fornonlinear systems

Nataliya Sedova1

1 Ulyanovsk State University, Ulyanovsk, Russia; [email protected]

Some controlled systems (in particular, non-holonomic systems) can-not be stabilized by continuous static feedback at the origin. Stabilizing

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feedback laws have been found for several such systems using either time-varying feedback or dynamic feedback. Another approach is to construct apiecewise continuous feedback laws, which are widely used in the optimalcontrol theory also.

In this paper piecewise continuous control laws for nonlinear time-invariant systems are considered. Piecewise smooth Lyapunov functionsor control Lyapunov functions can be used to construct the control lawand to study behavior of trajectories of the controlled system.

Let Rn be the real n-vector space with a norm | · |, R+ = [0,+∞),x = x(t) be the right derivative at t of a function x : R+ → Rn, andx(t, x0) denote the solution of a differential equation with x(0) = x0.

Let G ⊂ Rn be a connected open set, 0 ∈ G, f : (0 ∪ G) → Rn,f(0) = 0. Also, let M be an open subset of Rn such that 0 ∈ M ⊂(0 ∪ G). We say that origin 0 ∈ Rn is asymptotically stable for thesystem x = f(x) on the set M , if for any δ > 0 there exists ε > 0 suchthat for x0 ∈M ∩ |x| < δ we have x(t, x0) ∈M ∩ |x| < ε for all t > 0and |x(t, x0)| → 0 as t → +∞ (in particular, M is f -invariant, i.e. everytrajectory starting at x0 ∈ M remains in M for all t > 0).

Let G1, G2 be open connected subsets of Rn such that G1 ∪ G2 =Rn \ 0, f i ∈ C(Gi, R

n), i = 1, 2. Also assume that there exists aseparating hypersurface S with 0 ∈ S and S \ 0 ⊂ G1 ∩G2.

Let M1, M2 be connected components of Rn \ S, Mi ⊂ Gi, and fipoints towards Mi on S for i = 1, 2. If the the origin is asymptoticallystable for x = f1(x) on G1 and for x = f2(x) on G2, then the zero solutionof the system

x =

f1(x) if x ∈ (S \ 0) ∩M1

f2(x) if x ∈M2

0 if x = 0(1)

is asymptotically stable [1].Sufficient conditions of asymptotic stability for system (1) can be

stated in terms of two Lyapunov functions V1 and V2 defined on G1 andG2 respectively. This idea can be applied to controlled systems of theform x = f(x) + g(x)u with an input u ∈ Rm: two control Lyapunovfunctions Vi : Gi → R+, Vi ∈ C1(Gi, R

+), Vi(0) = 0, and the stabilizingfeedback control of the form

u =

u1(x) if x ∈ S ∩M1

u2(x) if x ∈M2

228

can be constructed.We also consider systems of the form

x =

f1(x) if x ∈M1

f2(x) if x ∈M2

αf1(x) + (1− α)f2(x) if x ∈ S(2)

without invariance assumption. There are three basic types of solutionsbehavior along S: a transversal motion with solution passing from oneregion Mi to another, the sliding mode, and the case of non-uniqueness.Lyapunov functions for such systems are defined and some stability andattraction results for the system (2) are stated. Some applications of theresults to the optimal control problem are given.


(project no. 11-01-00541).

References

1. G.A. Lafferriere, E.D. Sontag. “Remarks on Control Lyapunov Functions forDiscontinuous Stabilizing Feedback,” In Proc. IEEE Conf. Decision andControl, San Antonio, Dec. 1993, IEEE Publications, 306–308 (1993).

Optimal control of nonlinear parabolic equations andthe differentiability of the control-state mapping

Simon Serovajsky1

1 al-Farabi Kazakh National University, Almaty, Kazakhstan

[email protected]

Practical methods for solving optimization problems used, as a rule,necessary conditions of optimality or gradient methods. These methodsrequire finding the derivative of the minimizing functional. Therefore wehave the necessity to prove the differentiability of the control-state map-ping. This property can be substantiated with using Inverse Function

229

Theorem or Implicit Function Theorem. These results are true when-ever the derivative of the state operator is invertible. This assumptionand analogues of Lustenik’s condition use in known methods of the gen-eral extremum theory. Unfortunately these suppositions are broken forthe large class of nonlinear infinite dimensional systems. Moreover thecontrol-state mapping can be non-differentiable in this case. It is truewithout any nonsmooth terms in the cost functional and the state equa-tion. So the well-known methods of nonsmooth optimization are inappli-cable for these problems. We will overcome these difficulties by means ofthe extended differentiation theory.

Let Ω be an open bound set of the space Rn with boundary Γ, T > 0,Q = Ω× (0, T ), Σ = Γ× (0, T ). Consider nonlinear parabolic equation

y′ −∆y + |y|ρy = vQ + fQ, (x, t) ∈ Q (1)

with boundary conditions

y = 0, (x, t) ∈ Σ, (2)

y(x, 0) = vΩ + fΩ, x ∈ Ω, (3)

where y′ = ∂y/∂t, ρ > 0. DetermineW = L2

(0, T ;H1

0(Ω))and its adjoint

space W ′ = L2

(0, T ;H−1(Ω)

). Consider the spaces X = W ∩ Lq(Q) and

X ′ = W ′ + Lq′(Q), where q = ρ + 2, 1/q + 1/q′ = 1. The functionsfQ ∈ X ′ and fΩ ∈ L2(Ω) are known. The pair v = (vQ, vΩ) from the spaceV = L2(Q)×L2(Ω) is chosen as a control. By monotone operators theory(see [1], Chapter VI, Theorem 1.1), for any v ∈ V the problem (1) – (3)has a unique solution y = y[v] from the space Y =

y| y ∈ W, y′ ∈ W ′,

besides the map y[.] : V → Y is weakly continuous.Consider the functional

I(v, y) =

∫

Q

FQ(ξ, y(ξ),∇y(ξ), vQ(ξ)

)dQ +

∫

Ω

FΩ

(x, y(x, T ), vΩ(x)

)dΩ,

where FQ and FΩ are known functions. It is given the set of admissiblecontrols U = UQ × UΩ, where UQ and UΩ are nonempty convex closedbound subsets of the spaces L2(Q) and L2(Ω). Determine the functionalJ(v) = I(v, y[v]).

P r o b l e m P . Minimize the functional J on the set U .

230

The solvability of this problem is guaranteed by the following result.

Theorem 1. Suppose FQ : Q× Rn+2 → R and FΩ : Ω× R

2 → R areCaratheodory functions, which satisfy inequalities

FQ(ξ, ϕ, ψ) ≥ Φ(|ψ|) ∀ξ ∈ Q, ϕ ∈ R, ψ ∈ Rn+1,

FΩ(x, η) ≥ Ψ(|η|) ∀x ∈ Ω, η ∈ R2,

where Φ : R+ → R+ and Ψ : R+ → R+ are increasing convex coercivesemicontinuous functions; besides FQ(ξ, ϕ, .) and FΩ(x, .) are convex onthe sets R

n+2 and R2 for all ξ ∈ Q, ϕ ∈ R, x ∈ Ω. Then Problem P is

solvable.

The necessary condition of the minimum of the functional J on theset U at the point u is the variational inequality

J ′(u)(v − u) ≥ 0 ∀v ∈ U, (4)

where J ′(u) is the derivative of J at the point u. The application of thisresult for our problem requires the justification of the differentiability ofthe control-state mapping. However we have the following assertion.

Lemma 1. The map y[.] :(X ′×L2(Ω)

)→ Y is not Gateaux differen-

tiable for large enough ρ and n.

Thus necessary conditions of the optimality for Problem P can beobtained only for small enough values of the nonlinearity parameter ρand the dimension n of the set. The differentiability of the control-statemapping can be proved actually with using of Inverse Function Theoremfor this case. Note that the known results for optimization control systemsdescribed by nonlinear parabolic equations use small enough values of thenonlinearity parameter and the dimension of the set only (see for example,[2], Chapter IV, Theorem 2.6; [3], Chapter 1, Theorem 3.2; [4], Chapter2, Theorem 8.1). However by Theorem 1, our problem is solvable forall values of these parameters. So it will be interesting to analyze itwithout any parameters constraints. The desirable result can be obtainedby means of an extension of operator derivative [5].

D e f i n i t i o n. An operator L : V → Y , where V and Y arelinear topological spaces, is called (V0, Y0;V∗, Y∗)-extended differentiable

231

at a point u ∈ V , if there exist linear topological spaces V0, Y0, V∗, Y∗with continuous embeddings V∗ ⊂ V0 ⊂ V , Y ⊂ Y0 ⊂ Y∗ and a linearcontinuous operator D : V0 → Y0 such that

[L(u + σh) − Lu

]/σ → Dh

in Y∗ for all h ∈ V∗ as σ → 0.

For any point u ∈ V determine the spaces

X0(u) =y∣∣ y ∈W,

∣∣y[u]∣∣ρy ∈ L2(Q)

,

X ′0(u) =

µ+

∣∣y[u]∣∣ρ/2η

∣∣ µ ∈W ′, η ∈ L2(Q),

Y0(u) =y∣∣ y ∈ X0(u) y

′ ∈ X ′0(u)

.

Lemma 2. The map y[.] : V → Y is (V, Y0(u);V,W )-extended differ-entiable at the arbitrary point u ∈ V . Its extended derivative y′[u] satisfiesthe equality ∫

Q

µQy′[u]hdQ +

∫

Ω

µΩ

(y′[u]h

)|t=TdΩ

=

∫

Q

p[µ]hdQ +

∫

Ω

p[µ]|t=0hΩdΩ ∀µ ∈(X ′

0(u)× L2(Q)),

where µ =(µQ, µΩ

), p[µ] is the solution of the boundary problem

−p[µ]′ −∆p[µ] + (ρ+ 1)|y[u]|ρp[µ] = µQ, (x, t) ∈ Q,p[µ] = 0, (x, t) ∈ Σ,

p[µ](x, T ) = µΩ, x ∈ Ω.

Besides(y[u+σh]|t=T −y[u]|t=T

/σ →

(y′[u]h

)|t=T in L2(Ω) as σ → 0.

Then we can prove the differentiability of the minimizing functional.

Lemma 3. Suppose the assumptions of Theorem 1 are true, andthe functions FQ(ξ, .) and FΩ(x, .) have continuous derivatives FQ0,...,FQ(n+1) and FΩ1, FΩ2; besides

∣∣FQ(ξ, η)∣∣≤ aQ(ξ) + bQ

n+1∑

i=0

|ηi|2,∣∣FQj(ξ, η)

∣∣≤ aQj(ξ) + bQj

n+1∑

i=0

|ηi|,

232

∣∣FΩ(x, ζ)∣∣≤ aΩ(x)+ bΩ

(|ζ1|2+ |ζ2|2

),∣∣FΩ;(x, ζ)

∣∣≤ aΩl(x)+ bΩl(|ζ1|+ |ζ2|

)

for all η ∈ Rn+2, ζ ∈ R

2, ξ ∈ Q, x ∈ Ω, where aQ ∈ L1(Q), aQj ∈ L2(Q),aΩ ∈ L1(Ω),aΩl ∈ L2(Ω),bQ > 0, bQj > 0,bΩ > 0, bΩl > 0, j = 0, ..., n+ 1,l = 1, 2. Then the functional J has Gateaux derivative at the arbitrarypoint u ∈ V such that

J ′(u)h =

∫

Q

(FQu + p)hQdQ +

∫

Ω

(FΩu + p|t=T

)hΩdΩ ∀h ∈ V,

where p is the solution of the adjoint system

−p′ −∆p+ (ρ+ 1)|y[u]|ρp = FQy − divFQ∇y , (x, t) ∈ Q,

p = 0, (x, t) ∈ Σ,

p(x, T ) = FΩy , x ∈ Ω.

BesidesFQy(ξ) = FQ0

(ξ, y[u](ξ), y[u](ξ), uQ(ξ)

),

FQ∇y(ξ) =(FQi

(ξ, y[u](ξ), y[u](ξ), uQ(ξ)

)), i = 1, ..., n,

FQu(ξ) = FQ(n+1)

(ξ, y[u](ξ), y[u](ξ), uQ(ξ)

),

FΩy(x) = FΩ1

(x, y[u](x, T ), uΩ(x)

).

FΩu(x) = FΩ2

(x, y[u](x, T ), uΩ(x)

).

Using Lemma 3 and the formula (4), we get necessary conditions ofoptimality.

Theorem 2. Under the conditions of Lemma 3, the optimal controlu = (uQ, uΩ) satisfies the variational inequalities

∫

Q

(FQ + p

))(vQ − uQ

)dQ ≥ 0 ∀vQ ∈ UQ,

∫

Ω

(FΩ + p|t=T

))(vΩ − uΩ

)dΩ ≥ 0 ∀vΩ ∈ UΩ.

233

Thus necessary conditions of optimality for Problem P involve thestate system (1) – (3) for v = u, adjoint system and variational inequali-ties.

References

1. H. Gajewski, K. Groger, K. Zacharias Nichtlineare Operatorgleichungen undOperatordifferentiagleichungen, Academie Verlag, Berlin (1974).

2. P. Neittaanmaki, D. Tiba Optimal Control of Nonlinear Parabolic Systems,Theory, Algorithms, and Applications, Marcel Dekker, New York (1994).

3. J.L. Lions Controle de Systemes Distribuas Singuliers, Gauthier-Villars, Paris(1983).

4. A.V. Fursikov Optimal Control of Distributed Systems. Theory and Appli-cations, Amer. Math. Soc., Providence (1999).

5. S.Ya. Serovajsky “Calculation of functional gradients and extended differen-tiation of operators,” J. Inverse Ill-Posed Problems, 13, No 4, 383–396 (2005).

Inverse problems for parabolic equations with infinitehorizon

Iliyas Shakenov1

1 al-Farabi Kazakh National University, Almaty, Kazakhstan;

[email protected]

Practical problems often lead to difficulty that we call inverse prob-lems. For example, if you need to know the temperature of soil at thedepth of several meters while it is possible to measure the temperatureonly on the surface. In this type of problems there is a lot of information(over-determination) on one side of the boundary but no any data at theother side.

We consider one of such problems with the following mathematicalproblem definition.

Let we are given a one-dimensional heat conduction equation and

234

initial-boundary problem

ut = uxx + f(x, t), 0 < x < L, 0 < t <∞, (1)

u|t=0 = ϕ(x), (2)

ux|x=0 = b(t), (3)

ux|x=L = y(t). (4)

Right boundary function y(t) is unknown and has a meaning of heatflow. To determine this value we can use an additional informationu(0, t) = a(t). To solve the problem numerically we cannot use infinitetime interval, so we replace the original problem with its finite approxima-tion. That is, 0 < t < T , where T becomes larger and larger. We convertthis problem to optimization, in which we are required to minimize thefunctional on (1) – (4):

I(y) =

T∫

0

(u(0, t; y)− a(t)

)2dt→ min

If we find control y(t) which gives a zero value to the functional thenthe additional information is fulfilled.

We use one familiar method to solve the problem by constructing aniterative process.

yn+1(t) = yn(t)− αnI ′(yn(t)

),

where αn > 0.The first problem is to determine what is I ′ and how to find it? Using

an apparatus of Gato derivative we derive the theorem, [1]:

Theorem. The Gato derivative of functional I in point y(t) is equalto ψ(L, t), where ψ(x, t) is a solution of adjoint problem

ψt + ψxx = 0, (5)

ψ(x, T ) = 0, (6)

ψx(L, t) = 0, (7)

235

ψx(0, t) = −2(u(0, t; y)− a(t)

). (8)

Solution algorithm.1. Parameters setting. Such as deviation of functional from zero,

step by time and step by x-axis, initialization of known functions. Chooseinitial approximation y0(t).

2. Direct problem solution. We solve (1) – (4) and get u(0, t; y).3. Calculation of the functional. If I(y) < ε, then the algorithm

terminates and results are shown. If I(y) > ε, then go to item 4.4. Adjoint problem solution. Here we get I ′(y) = ψ(L, t).5. We choose α–value. It might be a constant or it can be calculated

by specific formulas, [2].6. Next approximation construction. We use the mentioned

formula:yn+1(t) = yn(t)− αnI ′

(yn(t)

).

Go to item 2.Detailed description of all algorithm steps are given in the work, [3].Once we have constructed a computational algorithm, we can do a lot

of experiments (accompanied with graphs and tables) and draw conclu-sions about the following issues:

1. First of all, having in hand exact solution, we can compare it withthe derived by computational algorithm.

2. We analyze an influence of steps by time and x-axis, closeness offunctional to zero on the solution of the problem.

3. The inverse problems like this has an effect of insolvability on theright-hand side of time-interval. We consider how it might be resolvedand of course we touch the problem about in what fraction of entire time-interval we can get sufficient accuracy.

4. Different ways to chose parameter α has been applied and com-pared. Some of them – traditional methods – turned out very inefficient.

5. We elongate the time-interval and observe solutions behavior. Alsowe examine some more methods of constructing successive approximationsyn+1(t) when time increases and compare them with each other. Themain idea is in calculating the solution on small interval and filling therest values with zero. This solution is used as initial approximation whentime interval becomes larger.

236

6. Attempts to transform an equation using small parameter are doneand the results compared with original method.

7. Theoretical researches about convexity of the functional, unique-ness of the solution, applicability of the maximum principle are also sub-mitted.

Additionally we give an outline of the future works to improve anddevelop this research, which we believe has a great applicability and con-cerns about new unknown effects in inverse problem theory.

References

1. S.Y. Serovajsky Optimization and differentiation, Print-S, Almaty, (2006).(In Russian)

2. S.I. Kabanihin Inverse and ill-conditioned problem, Novosibirsk, (2009). P.349 – 354. (In Russian)

3. A.A. Samarsky Theory of difference schemes, Nauka, oscow, (1983).(In Rus-sian)

Solution of the parametric inverse problem ofstochastic optimal control

Kanat Shakenov1

1 al-Farabi Kazakh National University, Almaty, Kazakhstan;

[email protected]

Problem definition.According to [1] we consider a problem of estimation of unknown den-

sity parameter when additional control parameter presents and can bechosen randomly. Let f

(y|x, z

)is a distribution density with respect to

measure µ(dy) on a line with dependence on parameters x and z. Let as-sume that x, y, z are one–dimensional: x ∈ X ⊂ E1, y ∈ E1, z ∈ Z ⊂ E1,where E1 is Euclid space with dimension of 1. Parameter x = x0 isunknown and must be estimated by means of samples with density f .Parameter z has a meaning of control and it can be chosen in two ways:

237

first, it can be constant; second, i-th observation implies the followingvalue:

Zi = Zi(Y1, . . . , Yi−1

)= Zi

(Y (i−1)

). (1)

More precisely about (1): simultaneous distribution of Y1, . . . , Yn can bedetermined by formula

fn

(Y(n), x0

)=

n∏

i=1

f(yi|x0, zi

(Y(i−1)

)),

where Y(n) =(y1, . . . , yn

).

We shall denote Zi =(Z1, . . . , Zi

)and call it as acceptable control or

plan of experiment or just plan if a sequence Z1, Z2, . . . satisfies (1) andsuch that Zi ∈ Z, i = 1, 2, . . . . The problem is to chose such a control(plan) which provides the best quality of estimation in sense of a given

quality criteria. We take this criteria as E(xn − x0

)2for a while.

Solution existence. Problems.

Let for all x ∈ X, z ∈ Z information amount I(x, z) =∫ ( ∂f(y|x,z)

∂x

)2f(y|x,z) ν(dy)

of density f exists, where f(y|x, z) is a PDF with respect to measure

ν(dy) on line ( x and z are parameters). Assume that Ex(xn − x

)2 ≥(Ex

n∑i=1

I(x, Zi

(Y (i−1)

)))−1

and Isup(x) = supz∈Z

I(x, z) < ∞, x ∈ X,

where Zi = Zi(Y1, . . . , Yi−1

)= Zi

(Y (i−1)

)is a sample. It is proven that

for any experiment plan the following inequality is true Ex(xn − x

)2 ≥1

nIsup(x)and asymptotically optimal plan Z1, . . . , Zn, . . . exists. For this

sequence of controls there is a sequence of estimations xn = xn(Y (n)

)such

that√n(xn − x) ∼ N

(0, 1

Isup(x0)

), n → ∞. Indicated problem can have

an exact solution if estimating parameter has probability distributions.Solution of such a problem lies in the theory of statistical solutions. (Seefor [2] or [3] and others). But this solution not always good for practice.We need in a priori distribution. It’s a difficult problem if a sample sizen is small. But for our case we can neglect it as we construct estimationsfor n→∞. For quite common assumptions Bayesian estimation with anyprior density π(x),

(π(x) 6= 0, x ∈ X

), is asymptotically optimal if x0

is not random value. You can see [4] for this purposes. There is a muchmore difficult problem. If you want to obtain more precise solution, it will

238

take a lot of numerical computations. This problem is considered in [3],chapter IV.

Problem of estimating with independent samples.Let’s consider a problem of estimation without control. Let Y1, Y2, . . . , Yn

are independent random vectors (sample) from El, each of which has adistribution density of f(y, x) with respect to σ-finite measure ν(·). Pa-rameter x takes values from open set X ∈ Ek.

Problems.By values of Y1, Y2, . . . , Yn and f(y, x) find the value of x. We have

no any prior information about x, except of x ∈ X. We try to construct

an estimation xn

(Y1, Y2, . . . , Yn

)of x. Nn is any measurable random

value, where Nn is minimal σ-algebra of events, and all the values ofY1, Y2, . . . , Yn are measurable with respect to it.

Let for all x ∈ X, z ∈ Z information amount I(x, z) =∫ ( ∂f(y|x,z)

∂x

)2f(y|x,z) ν(dy)

of density f exists, where f(y|x, z) is a PDF with respect to measure ν(dy)

on line ( x and z are parameters), [5]. Assume that Ex(xn − x

)2 ≥(Ex

n∑i=1

I(x, Zi

(Y (i−1)

)))−1

and Isup(x) = supz∈Z

I(x, z) < ∞, x ∈ X,

where Zi = Zi(Y1, . . . , Yi−1

)= Zi

(Y (i−1)

)is a sample. It is proven that

for any experiment plan the following inequality is true Ex(xn − x

)2 ≥1

nIsup(x)and asymptotically optimal plan Z1, . . . , Zn, . . . exists. For this

sequence of controls there is a sequence of estimations xn = xn(Y (n)

)

such that√n(xn − x) ∼ N

(0, 1

Isup(x0)

), n→∞.

References

1. M.B. Nevelson, R.Z. Hasminsky The stochastic approximations and recurrentestimating. Nauka, Moscow, (1972). (In Russian)

.

2. D. Blackwell, M.A. Girshick The theory gambles of stochastic solutions. IL.Moscow, (1958). (In Russian)

.

3. A.A. Feldbaum The foundational theory of optimal automatics system.Nauka, Moscow, (1966). (In Russian)

.

4. I.A. Ibragimov, R.Z. Hasminsky The asymptotic conducts of certain statis-tical estimations. Journal ”The Probability Theory and Applications”, 17, 3(1972). (In Russian)

.

5. H. Robbins , S. Monro A stochastic approximation method. Ann. Math.Statist., 22, 1 (1951), P. 400 – 407

.

239

Graph-based local elimination algorithms for sparsediscrete optimization problems

O. Shcherbina1, A. Sviridenko2

1 University of Vienna, Vienna, Austria; [email protected] Tavrian National University, Simferopol, Ukraine; [email protected]

1. IntroductionThe use of discrete optimization (DO) models and algorithms makes it

possible to solve many practical problems, since the discrete optimizationmodels correctly represent the nonlinear dependence, indivisibility of anobjects, consider the limitations of logical type and all sorts of technologyrequirements, including those that have qualitative character. To meetthe challenge of solving large scale DO problems (DOPs) in reasonabletime, there is an urgent need to develop and study new decompositionapproaches. Among decomposition approaches appropriate for solvingsparse DO problems we mention local elimination algorithms (LEA) [1],[2], which can exploit sparsity in the interaction graph of a DOP and allowto compute a solution in stages such that each of them uses results fromprevious stages. LEAs compute global information using local computa-tions (i.e., computations of information about elements of neighborhoodsof variables or constraints - usually, solving subproblems).

The algorithmic scheme of LEA is defined by an elimination tree [3]whose vertices are associated with subproblems and whose edges expressinformation on interdependence between subproblems. The structure ofa DO problem can be defined either by an interaction graph of initialelements (variables and constraints), or by various derived structures, e.g.,block structures, block-tree structures defined by a so called condensedgraph.

There are various computational schemes for realizing LEA, includingthe LEA elimination of variables, block-elimination algorithm, LEA basedon tree decomposition.

2. Block local elimination scheme2.1. Partitions, clustering, and quotient graphs

Consider the integer linear programming (ILP) problem with binaryvariables

240

f(X) = CX =

n∑

j=1

cjxj → max

subject to constraints

n∑

j=1

aijxj ≤ bi, i = 1, 2, . . . ,m,

xj = 0, 1, j = 1, 2, . . . , n.

The local elimination procedure can be applied to elimination of not onlyseparate variables but also to sets of variables and can use the so calledelimination of variables in blocks, which allows to eliminate several vari-ables in block.

Applying the method of merging variables into meta-variables allowsto obtain condensed or meta-DOPs which have a simpler structure. If theresulting meta-DOP has a nice structure (e.g., a tree structure) then itcan be solved efficiently.

An ordered partition of a set X is a decomposition of X into orderedsequence of pairwise disjoint nonempty subsets whose union is all of X .

In general, graph partitioning is NP -hard. Since graph partitioning isdifficult in general, there is a need for approximation algorithms. A popu-lar algorithm in this respect is MeTiS1, which has a good implementationavailable in the public domain.

An important special case of partitions are so-called blocks. Two vari-ables are indistinguishable if they have the same closed neighborhood. Ablock is a maximal set of indistinguishable vertices. The blocks of G par-tition X since indistinguishability is an equivalence relation defined onthe original vertices. The corresponding graph is called condensed graph,which is a merged form of original graph.

An equivalence relation on a set induces a partition on it, and also anypartition induces an equivalence relation. Given a graph G = (X,E), letX be a partition on the vertex set X : X = x1, x2, . . . , xp, p ≤ n, wherexl = XKl

(Kl is a set of indices corresponding to xl, l = 1, . . . , p). For thisordered partition X , the DOP (1) – (3) can be solved by the LEA usingquotient interaction graph G.

1http://www-users.cs.umn.edu/∼karypis/metis

241

That is, ∪pi=1xi = X and xi∩xk = ∅ for i 6= k. We define the quotientgraph of G with respect to the partition X to be the graph

G = G/X = (X, E),

where (xi, xk) ∈ E if and only if NbG(xi) ∩ xk 6= ∅.The quotient graph G(X, E) is an equivalent representation of the

interaction graphG(X,E), whereX is a set of blocks (or indistinguishablesets of vertices), and E ⊆ X × X be the edges defined on X. A localblock elimination scheme is one in which the vertices of each block areeliminated contiguously. As an application of a clustering technique weconsider below a block local elimination procedure where the eliminationof the block (i.e., a subset of variables) can be seen as the merging of itsvariables into a meta-variable.

2.2. Block local elimination algorithmA. Forward part

Consider first the block x1. Then

maxXCNXN |AiSi

XSi≤ bi, i ∈M, xj = 0, 1, j ∈ N =

maxXK2 ,...,XKp

CN−K1XN−K1 + h1(Nb(XK1)|AiSiXSi≤ bi, i ∈M − U1,

xj = 0, 1, j ∈ N −K1where U1 = i : Si ∩K1 6= ∅ and

h1(Nb(XK1)) = maxXK1

CK1XK1 |AiSiXSi≤ bi, i ∈ U1, xj = 0, 1, xj ∈ Nb[x1].

The first step of the local block elimination procedure consists of solving,using complete enumeration of XK1 , the following optimization problem

h1(Nb(XK1)) = maxXK1

CK1XK1 |AiSiXSi≤ bi, i ∈ U1, xj = 0, 1, xj ∈ Nb[x1],

and storing the optimal local solutions XK1 as a function of the neighbor-hood ofXK1 , i.e., X

∗K1

(Nb(XK1)).The maximization of f(X) over all feasible assignments Nb(XK1), is

called the elimination of the block (or meta-variable) XK1 . The optimiza-tion problem left after the elimination of XK1 is:

maxX−XK1

CN−K1XN−K1 + h1(Nb(XK1))|AiSiXSi≤ bi, i ∈M − U1,

242

xj = 0, 1, j ∈ N −K1.Note that it has the same form as the original problem, and the tabu-

lar function h1(Nb(XK1)) may be considered as a new component of themodified objective function. Subsequently, the same procedure may be ap-plied to the elimination of the blocks – meta-variables x2 = XK2 , . . . ,xp =XKp

, in turn. At each step j the new component hxjand optimal local

solutions X∗Kj

are stored as functions of Nb(XKj| XK1 , . . . , XKj−1), i.e.,

the set of variables interacting with at least one variable of XKjin the

current problem, obtained from the original problem by the eliminationof XK1 , . . . , XKj−1 . Since the set Nb(XKp

| XK1 , . . . , XKp−1) is empty,the elimination of XKp

yields the optimal value of objective f(X).B. Backward part.This part of the procedure consists of the consecutive choice of X∗

Kp,

X∗Kp−1

, . . . , X∗K1

, i.e., the optimal local solutions from the stored tables

X∗K1

(Nb(XK1)), X∗K2

(Nb(XK2 | XK1)), . . . , X∗Kp| XKp−1 , . . . , XK1 .

Underlying DAG of the local block elimination procedure containsnodes corresponding to computing of functions hxi

(NbG

(i−1)X

(xi)) and is

a generalized elimination tree.3. Comparative computational experimentAmong extremely important research questions about the effectiveness

of local elimination algorithms (LEA), the next one causes special interest:“Is the use of LEA in combination with a discrete optimization (DO)algorithm (for solving problems in the blocks) consistently more efficientthan the standalone use of the DO algorithm?” [4].

The computational capabilities of the LEA in combination with amodern solver were tested by using SYMPHONY2 as the implementa-tion framework. SYMPHONY is part of the COIN-OR3 project and itcan solve mixed-integer linear programs (MILP) sequentially or in paral-lel. We chose this framework since it is open-source and supports warmrestarts, which implement postoptimal analysis (PA) of ILP problems.

All experimental results were obtained on an Intel Core 2 Duo at 2.66GHz machine with 2 GB main memory, and running Linux, version 2.6.35-24-generic. SYMPHONY 5.4.14 was used for the LEA implementation.The maximum solving time is denoted by TIMEOUT , and is equal to 2

2https://projects.coin-or.org/SYMPHONY3http://www.coin-or.org4http://www.coin-or.org/download/source/SYMPHONY/

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hours. All the ILP problems with binary variables from a given experimenthave artificially generated quasi-block structures. All the blocks from asingle problem have the same number of variables, and also the samenumber of variables in separators between them. This is required in orderto evaluate the impact of the PA on the time to solve the problem byincreasing the number of variables.

The test problems were generated by specifying the number of vari-ables, the number of constraints and the size of the separators betweenblocks. The objective function and constraint matrix coefficients, andthe right-hand sides for each of the block were generated by using apseudorandom-number generator.

Each test problem was solved by using three algorithms, a) the basicMILP SYMPHONY solver with the OsiSym interface, b) the LEA incombination with SYMPHONY, c) the LEA in combination with SYM-PHONY and with PA (warm restarts). In all the cases SYMPHONY usedpreprocessing.

The computational experiments are described in details in [5] and showthat LEA combined with SYMPHONY for solving quasi-block problemswith small separators outperforms the stand alone SYMPHONY solver.Additionally, by increasing the size of the separators in the problems forthe same number of variables and block sizes LEA becomes less efficientdue to the increased number of iteration for solving the block subproblems.

References

1. Y.I.Zhuravlev. “Local algorithms of information computation,” I,II. Kiber-netika, 1, 12–19 (1965); 2, 1–11 (1966).

2. O. Shcherbina. “Graph-based local elimination algorithms in discrete opti-mization,” in: Foundations of Computational Intelligence Volume 3. GlobalOptimization Series: Studies in Computational Intelligence, Vol. 203,Springer, Berlin/Heidelberg, 2009, pp. 235–266.

3. J.W.H Liu. “The role of elimination trees in sparse factorization,” SIAM J.on Matrix Analysis and Applications, 11, 134–172 (1990).

4. O. Shcherbina. “Local Elimination Algorithms for Sparse Discrete Op-timization Problems,” D.Sc. Thesis, Computer Centre of RAS, Moscow(2011)

.

5. A. Sviridenko, O. Shcherbina. “Block local elimination algorithmsfor solving sparse discrete optimization problems,” available online:http://arxiv.org/abs/1112.6335 (2011)

.

244

Heuristic Algorithms for a Job-Shop Problem withMinimizing Total Job Tardiness

Yuri N. Sotskov1, Omid Gholami1, Frank Werner2

1 United Institute of Informatics Problems, 220012 Minsk, Belarus;

[email protected], gholami [email protected] Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, PSF

4120, 39016 Magdeburg, Germany; [email protected]

In practice, it is often required to process a set of jobs without opera-tion preemptions satisfying temporal and resource constraints. Temporalconstraints say that some jobs have to be finished before some otherscan be started. Resource constraints say that operations processed onthe same machine cannot be processed simultaneously. The objective isto construct a schedule specifying when each operation starts such thatboth temporal and resource constraints are satisfied and the given objec-tive function has a minimum value. One can model such a schedulingprocess via the following job-shop problem.

There are n jobs J = J1, J2, . . . , Jn to be processed on m machinesM = M1,M2, . . . ,Mm. Operation preemptions are not allowed, and themachine routes Oi = (Oi1, Oi2, . . . , Oini

) for processing the jobs Ji ∈ Jmay be given differently for different jobs. The time pij > 0 needed forprocessing an operation Oij of a job Ji on the corresponding machineMv ∈ M is known before scheduling. A job Ji ∈ J is available forprocessing from time-point ri ≥ 0. The time-point di > ri defines a duedate for completing job Ji. It is assumed that machine Mk ∈ M canprocess a job Ji ∈ J at most once. Consequently, any two operations Oijand Oik, j 6= k, of the same job Ji ∈ J have to be processed by differentmachines and ni ≤ m (such a scheduling problem is called a classical job-shop). We consider the objective of finding a schedule minimizing the sum∑n

i=1 Ti of the tardiness times Ti = max0, Ci − di for the jobs Ji ∈ J .Hereafter, Ci denotes the completion time of a job Ji ∈ J . According tothe three-field notation α|β|γ used for machine scheduling problems, theabove problem is denoted as J |ri|

∑Ti.

Problem J |ri|∑Ti arises, e.g., in train scheduling for a single-track

railway network: To determine a schedule for a set of trains that does notviolate the single-track capacities and the train timetable. In a single-

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track railway, a pair of sequential stations can be connected by a single-track (railroad section) only. Specifically, this is the case for most railwaynetworks in countries of the Middle East.

In a job-shop approach to train scheduling, trains and railroad sectionsare synonymous with the jobs J and the machines M, respectively. Anoperation Oij is regarded as a movement of the train Ji ∈ J across therailroad section Mv ∈ M, where machine Mv has to process operationOij . The positive number pij denotes the time required for train Ji ∈ Jto travel through section Mv ∈ M. The non-negative number ri denotesthe earliest possible departure time for the train (release time of the job)Ji ∈ J from the original station in the route Oi. The positive number didenotes the official arrival time of the train (due date for completing thejob) Ji ∈ J at the terminal station in the route Oi. It should be notedthat for train scheduling, the inequality m > n holds and each machineMk ∈M can process a job Ji ∈ J at most once.

Our aim was to find an algorithm for the problem J |ri|∑Ti to be fast

even for a large size of the input data (this is the case for a real-worldrailway scheduling problem). It is clear that an exact branch and boundmethod creates a lot of branches in the solution tree for a large inputdata, and so it is not possible to use a branch and bound method formost real-world job-shop problems with large sizes. Heuristic methodslike a genetic algorithm are basically rather slow. Furthermore, usingalgorithms like Lagrangian relaxation or simulated annealing can reducethe computational time only a bit. A lot of methods like tabu search needmany calculations and cannot satisfy our aim as well.

Problem J |ri|∑Ti is very complicated in the computational sense

since even its special cases belong to the class of binary (or unary) NP-hard problems [1]. In order to achieve a practical size of a classicaljob-shop problem, which can be solved heuristically within a reasonabletime, we first coded a shifting bottleneck algorithm, which was origi-nated in [2] for a job-shop problem J ||Cmax with the makespan criterionCmax = maxCi : Ji ∈ J . We tested the program realizing the shift-ing bottleneck algorithm for the problem J |ri|

∑Ti as one of the most

famous heuristic algorithms for job-shop problems [2] (this algorithm wasimproved in [3]). However, we obtained unsatisfactory large CPU-timesfor randomly generated instances J |ri|

∑Ti with large and even moderate

numbers m of machines provided that m > n. Therefore, we were forcedto look for other heuristic algorithms for the problem J |ri|

∑Ti, which

246

will run essentially faster than the shifting bottleneck algorithm and willprovide sufficiently close objective function values when m > n and eachmachine Mv ∈ M may process at most one operation from the route Oiof a job Ji ∈ J .

We observed that many recursive functions are needed to calculatedata like a local due date, a critical path and a local release time for eachvertex (i.e., operation) in the mixed graph G = (Q,A,E) representing ajob-shop [4]. In the mixed graph G = (Q,A,E), the vertex set Q is theset of all operations including a source operation O and a sink operationOi for each job Ji ∈ J . The arc set A defines temporal constraints givenby the routes Oi of the jobs Ji ∈ J . The edge set E defines resourceconstraints given by the machine setM = M1, M2, . . . ,Mm and by theroutes Oi for the jobs Ji ∈ J (see [4] for details).

The earliest start time rij of an operation Oij ∈ Q may be defined asthe length of the longest path from a start vertex O ∈ Q to vertex Oij inthe digraph (Q,A, ∅) obtained from the mixed graph G via deleting all theedges E. We call ri1 the release time of a job Ji provided that Oi1 ∈ Q isthe first operation of a job Ji in the route Oi. Since the feasible digraph(Q,A, ∅) has no circuits, all the earliest start times ri1, i ∈ 1, 2, . . . , n,are finite and can be calculated in linear time of the sum |Q|+ |A|.

We focus on the shortest release times of the operations Q in the fol-lowing algorithm which is called SRT-algorithm. The shortest release timeof an operation is used as a priority (SRT-priority) in the SRT-algorithm.In contrast to the shifting bottleneck algorithm, which examines a bot-tleneck machine at each iteration [2,3], a critical job is examined at eachiteration of the SRT-algorithm.

In the initial step of the SRT-algorithm, the earliest start times rij ofall operations Oij ∈ Q have to be computed due to the following recursion:rij = ri,j−1 + pi,j−1. The release time of the source operation O is equalto minri : Ji ∈ J .

The first job to be examined is a job Ji ∈ J , whose last operationOini

in the route Oi is the next to the last one in the critical path ofthe digraph (Q,A, ∅). At the first iteration of the SRT-algorithm, thefollowing two steps are realized.

Step 1. The SRT-algorithm finds the first request (i.e., operationOi1 ∈Q) of job Ji for the machine Mv ∈ M processing operation Oi1. Thenthe algorithm compares the release time ri1 with the release times ofall operations Ojg ∈ Q of the other jobs on the same machine Mv ∈ M

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processing operationOi1. To resolve conflicts of jobs for the same machine,the SRT-priority is used as follows. If the release time ri1 is not greaterthan that of operation Ojg for the same machine Mv ∈ M, then thearc (Oi1, Ojg) starting from operation Oi1 and ending in operation Ojghas to be added to the digraph (Q,A, ∅). Otherwise, the symmetric arc(Ojg , Oi1) has to be added to the digraph (Q,A, ∅). If an arc is added tothe digraph (Q,A, ∅), some local release times may be changed after thesecond step of the SRT-algorithm.

Step 2. The release time of the destination vertex of an arc and theother vertices related to this vertex must be checked and the correspondingrelease times must be modified if it is necessary. A job priority will bedefined by an arc between two operations requesting the same machine,one is the starting vertex of the arc (let this vertex be Okm) and the otherone is an end vertex of the arc (let it be Oij). A new release time of theoperation Oij has to be calculated as follows: rij := maxrij , rkm+pkm,where the maximum has to be taken over all arcs (Okm, Oij) belonging tothe digraph already constructed. The above equation must be recursivelyapplied to each vertex of the digraph that has an incoming arc from thevertex Oij until the sink vertex Oi. If the release time was not changed,the calculation of the recursive function is stopped.

Steps 1 and 2 are repeated for operation Oi2, then for operationOi3, and so on until operation Oini

of the route Oi. As a result, themixed graph G is transformed into a mixed graph denoted as Gi =(Q,A

⋃Ai, E \ Ei).

The second job to be examined is the “second critical” job, i.e., ajob Ju ∈ J \ Ji, whose last operation in the route Ou has the largestcompletion time in the digraph (Q,A

⋃Ai, ∅) among all jobs from the set

J \ Ji. So, at the second iteration, steps 1 and 2 are executed for thejob Ju, the digraph (Q,A

⋃Ai, ∅), and the mixed graph Gi.

The process is continued for the “third critical” job, then for the“fourth critical” job and so on, until all jobs J have been accounted.As a result, the mixed graph G is transformed into the digraph Gh =(Q,A

⋃Ah, E \ Eh), where E \ Eh = ∅ and job Jh was examined at the

h-th iteration. It is easy to convince that using the SRT-priority cannotgenerate a circuit in the digraphs (Q,A

⋃Ai, ∅), ..., (Q,A

⋃Ah, ∅) con-

structed from the first to the last iterations. A circuit-free digraph Ghuniquely determines a semiactive schedule [4], which may be built via thecritical path method in O(n+ |Ah|) time.

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Both the shifting bottleneck algorithm and the SRT-algorithm havebeen coded in Delphi and tested on a laptop computer. We comparedthe CPU-time taken by the SRT-algorithm and by the shifting bottle-neck algorithm to solve the same randomly generated problems J |ri|

∑Ti

heuristically for different values n ≤ 20 and m ≤ 20. The SRT-algorithmruns considerably faster than the shifting bottleneck algorithm if m > n.In the experiments, we compared the objective function values for ran-domly generated problems J |ri|

∑Ti with different products n×m.

The computational results showed that there is no meaningful differ-ence between the quality of the schedules obtained by the two algorithmstested for randomly generated problems J |ri|

∑Ti. The computational

experiments also evaluated the effect of adding either new jobs or newmachines to the CPU-time required to solve randomly generated prob-lems J |ri|

∑Ti. When we increased the number of jobs (machines, re-

spectively) of the randomly generated instances J |ri|∑Ti, the CPU-time

needed for the SRT-algorithm increased considerably (very slowly).The SRT-algorithm used a smaller CPU-time than the shifting bottle-

neck algorithm especially when the number of machines m was essentiallylarger than the number of jobs n. Since the total tardiness values

∑Ti

of the schedules constructed by the shifting bottleneck algorithm and theSRT-algorithm are rather close, we can claim that the SRT-algorithm isa good heuristic for a job-shop problem J |ri|

∑Ti with large m > n.

As observed from computational experiments, the SRT-algorithm is agood choice for the classical job-shop problem J |ri|

∑Ti when the number

of machines is much larger than the number of jobs.

References

1. P. Brucker, Yu.N. Sotskov and F. Werner, “Complexity of shop-schedulingproblems with fixed number of jobs: a survey,” Mathematical Methods ofOperations Research, Vol. 65, No. 3, 461 – 481 (2007).

2. J. Adams, E. Balas and D. Zawack, “The shifting bottleneck procedure forjobshop scheduling,” Management Science, Vol. 34, No. 3, 391 – 401 (1988).

3. E. Balas and A. Vazacopoulos, “Guided local search with shifting bottleneckjob shop scheduling,” Management Science, Vol. 44, No. 2, 262 – 275 (1998).

4. V.S. Tanaev, Yu.N. Sotskov and V.A. Strusevich, “Scheduling Theory: Multi-Stage Systems,” Kluwer Academic Publishers, Dordrecht, The Netherlands(1994).

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P -regular nonlinear optimization. High orderoptimality conditions

A.A. Tretyakov1,2,3

1 Dorodnicyn Computing Center RAS, Moscow, Russia;2 System Res. Inst., Polish Acad. Sie, Warsaw, Poland;

3 University of Podlasie, Siedlce, Poland;

[email protected]

Consider the following singular optimization problem

minφ(x), (1)

subject toF (x) = 0, (2)

where F : X → Y, X, Y — Banach spaces, and φ : X → R, F ∈ Cp+1(X),φ ∈ C2(X) and at the solution point x∗ we have

ImF ′(x∗) 6= Y. (3)

Elements of p-regularity theory

Let us recall the basic constructions of p-regularity theory which isused in solving of singular problems. The construction of the p-factor-operator. Suppose that the space Y is decomposed into a direct sum

Y = Y1 ⊕ · · · ⊕ Yp, (4)

where Y1 = ImF ′(x∗), Z1 = Y . Let Z2 be closed complementary subspaceto Y1 (we assume that such closed complement exists), and let PZ2 : Y →Z2 be the projection operator onto Z2 along Y1. By Y2 we mean theclosed linear span of the image of the quadratic map PZ2F

(2)(x∗)[·]2.More generally, define inductively,

Yi = span ImPZiF (i)(x∗)[·]i ⊆ Zi, i = 2, . . . , p− 1,

where Zi is a chosen closed complementary subspace for (Y1⊕ · · · ⊕ Yi−1)with respect to Y , i = 2, . . . , p, and PZi

: Y → Zi is the projection

250

operator onto Zi along (Y1 ⊕ · · · ⊕ Yi−1) with respect to Y , i = 2, . . . , p.Finally, Yp = Zp.

The order p is chosen as the minimum number for which (4) holds.Let us define the following mappings

Fi(x) = PYiF (x), Fi : X → Yi i = 1, . . . , p,

where PYi: Y → Yi is the projection operator onto Yi along (Y1 ⊕ · · · ⊕

Yi−1 ⊕ Yi+1 ⊕ · · · ⊕ Yp) with respect to Y , i = 1, . . . , p.

D e f i n i t i o n 1. The linear operator Ψp(h) ∈ L(X,Y1 ⊕ · · · ⊕ Yp),h ∈ X , h 6= 0,

Ψp(h) = F ′1(x

∗) + F ′′2 (x

∗)h+ · · ·+ F (p)p (x∗)[h]p−1,

is called the p-factor operator.

D e f i n i t i o n 2. We say that the mapping F is p-regular at x∗

along an element h, ifImΨp(h) = Y.

R e m a r k. The condition of p-regularity of the mapping F (x) at thepoint x∗ along h is equivalent to

F (p)p (x∗)[h]p−1 KerΨp−1(h) = Yp.

D e f i n i t i o n 3. We say that the mapping F is p-regular at x∗ ifit is p-regular along any h from the set

Hp(x∗) =

p⋂

k=1

KerkF(k)k (x∗)

\ 0,

where k-kernel of the k-order mapping F(k)k (x∗) is as follows

KerkF(k)k (x∗) = ξ ∈ X : F

(k)k (x∗)[ξ]k = 0.

For a linear surjective operator Ψp(h) : X 7→ Y between Banach spaceswe denote by Ψp(h)−1 its right inverse. Therefore Ψp(h)−1 : Y 7→ 2X

and we have

Ψp(h)−1(y) = x ∈ X : Ψp(h)x = y .

251

We define the norm of Ψp(h)−1 via the formula

‖Ψp(h)−1‖ = sup‖y‖=1

inf‖x‖ : x ∈ [Ψp(h)]−1(y).

We say that Ψp(h)−1 is bounded if ‖Ψp(h)−1‖ <∞.The following theorem gives a description of a solution set in degene-

rate case.

Theorem 1 (Generalized Lyusternik Theorem). Let X and Ybe Banach spaces and U be a neighborhood of x0 ∈ X. Assume thatF : X → Y , F ∈ Cp(U) is p-regular at x0. Then

T1M(x∗) = Hp(x∗).

Optimality conditions for p-regular optimization problems

We define p-factor Lagrange function

Lp(x, λ, h) = ϕ(x) +

(p∑

k=1

F(k−1)k (x)[h]k−1, λ

),

where λ ∈ Y ∗ and

Lp(x, λ, h) = ϕ(x) +

(p∑

k=1

2

k(k + 1)F

(k−1)k (x)[h]k−1, λ

).

D e f i n i t i o n 4. The mapping F is called strongly p-regular atthe point x∗ if there exists γ > 0 such that

suph∈Hγ

∥∥∥Ψp(h)−1∥∥∥ <∞,

where

Hγ =

h ∈ X :

∥∥∥F (k)k (x∗)[h]k

∥∥∥Yk

≤ γ, i = 1, p, ‖h‖ = 1

.

Let us recall the following basic theorems of the p-regularity theory.

Theorem 2 (Necessary and sufficient conditions for optima-lity). Let X and Y be Banach spaces, ϕ ∈ C2(X), F ∈ Cp+1(X), F :

252

X → Y , ϕ : X → R. Suppose that h ∈ Hp(x∗) and F is p-regular along

h at the point x∗. If x∗ is a local solution to the problem (1)–(2), thenthere exist multipliers, λ∗(h) ∈ Y ∗ such that

Lp′x(x∗, λ∗(h), h) = 0. (5)

Moreover, if F is strongly p-regular at x∗, there exist α > 0 and a multi-plier λ∗(h) such that (5) is fulfilled and

Lpxx(x∗, λ∗(h), h)[h]2 ≥ α‖h‖2 (6)

for every h ∈ Hp(x∗), then x∗ is a strict local minimizer to the problem

(1)–(2).

For our purposes will be useful the following modification of Theo-rem 1.

Theorem 3. Let X and Y be Banach spaces, ϕ ∈ C2(X), F ∈Cp+1(X), F : X → Y , ϕ : X → R. Suppose that h ∈ Hp(x

∗) and Fis p-regular along h at the point x∗. If x∗ is a local solution to the prob-lem (1)–(2), then there exist multipliers λi(h) ∈ Yi, i = 1, . . . , p, suchthat

ϕ′(x∗) + F ′(x∗)∗λ1(h) + · · ·+(F (p)(x∗)[h](p−1)

)∗λp(h) = 0 (7)

and (F (k)(x∗)[h](k−1)

)∗λi(h) = 0, i = 1, . . . , p \ k, (8)

k = 1, . . . , p.Moreover if f is strongly p-regular at x∗, there exist α > 0 and multi-

pliers λi(h), i = 1, . . . , p, such that (7)–(8) hold and

ϕ′′(x∗) +

[1

3F ′′(x∗)λ1(h) + · · ·+

+2

p(p+ 1)F (p+1)(x∗)[h]p−1λp(h)

][h]2 ≥ α‖h‖2

(9)

for every h ∈ Hp(x∗), then x∗ is a strict local minimizer to the prob-

lem (1)–(2).

The author were supported by the Russian Foundation for Basic Research

(grant No 11-01-00786a) and the Council for the State Support of Leading

Scientific Schools (grant 5264.2012.1).

253

Separating plane algorithm with additional clippingfor convex optimization

Evgeniya A. Vorontsova1

1 Institute of Automation and Control Processes, Far Eastern Branch of the

Russian Academy of Sciences; Far Eastern Federal University, Vladivostok,


We consider the solvable unconditional optimization problem

minx∈En

f(x) = f(x∗) (1)

where En – n-dimensional Euclidean space and f is a nonsmooth convexobjective function from En into E1.

There are many areas of applications for such algorithms because thiskind of problems occurs frequently in the industry. Moreover, the area oflarge-scale optimization will have more advantages of any impovementsin the methods for minimization of nondifferentiable functions.

Research in this direction has resulted in several efficient methods [1-5]. This paper presents an algorithm for solving the problem (1). Thealgorithm belongs to a class of separating plane algorithms [4-5].

The basic idea of separating plane algorithms (SPA) is to use the nextto trivial identity of convex analysis

minxf(x) = f(x∗) = −f∗(0)

where f∗(g) = supxxg − f(x) - the Fenchel-Moreau conjugate of the

function f .In this way the problem (1) can be reformulate as a problem of com-

puting f∗(0). The optimal point x∗ can be obtained as a subgradient off∗: x∗ ∈ ∂f∗(0).

The SPA algorithms construct sequences of outer and inner approx-imations of the epigraph of f∗ (epi f∗ = (ν, g) : ν ≥ f∗(g)). At eachiteration of the algorithm the approximations are gradually refined. Even-tually we obtain converging lower and upper bounds for f∗(0).

As the SPA algorithm [5] has no guarantee of monotony, the followingimprovement is suggested. At each iteration we execute an additional step

254

which removes the upper part of epi f∗:

sup(g, ǫ)∈ epi f∗; ǫ≤ v

gx− ǫ (2)

where estimate v is a solution to a problem v = min ǫ under the condition:(0, ǫ) ∈ co ((gk, f∗(gk)), k = 1, 2, ...) + 0×R+.

Using standard duality for (2), we reduce it to a line-search problem:

min0<θ≤1

1

θ(f(θx) + v) = min

0<θ≤1φ(θ). (3)

As f is a convex function, the function of a single variable φ(θ) is convex.It is suggested to solve the one-dimensional nonsmooth optimization

problem (3) by means of a new modification of line-search algorithm fornonsmooth convex optimization (see [6] for the original scheme).

Numerical experiments demonstrated quite satisfactory computationalperformance of the modified separating plane algorithm with additionalclipping.

The author was supported by the Far Eastern Branch of the Russian Academy

of Sciences (project no. 12-III-A-01N-014).

References

1. N.Z. Shor. Nondifferentiable Optimization and Polynomial Problems, KluwerAcademic Publishers, Dordrecht (1998).

2. J.-B. Hiriart-Urruty, C. Lemarechal. Convex Analysis and Minimization Al-gorithms II: Advanced theory and bundle methods. Fundamental Principlesof Mathematical Sciences 306, Springer-Verlag, Berlin (1993).

3. E. Polak, D.Q. Mayne. ”Algorithm models for nondiffentiable optimization,”SIAM J. Contr. Optimiz., 23, No. 3, 477–491 (1985).

4. E.A. Nurminski. Numerical Methods of Convex Optimization, Nauka,Moscow (1991).

5. E.A. Nurminski. ”Separating plane algorithms for convex optimization,”Mathematical Programming, 76, 373–391 (1997).

6. E.A. Nurminski. ”A quadratically convergent line-search algorithm for piece-wise smooth convex optimization,” Optimization Methods and Software, 6,59–80 (1995).

255

Equilibrium Model of the Russian Economy for theperiod of Global Financial Crisis

V.P. Vrzheshch1, N.P. Pilnik2, I.G. Pospelov3

1 Faculty of Computational Mathematics and Cybernetics MSU, Moscow,

Russia; [email protected] High School of Economy, Moscow, Russia; [email protected]

3 Dorodnicyn Computer Centre of RAS, Moscow, Russia; [email protected]

The report describes three product intertemporal equilibrium modelof the Russian economy. Three aggregated products: export, domesticand imported are distinguished by a special procedure of nonlinear dis-aggregation of macroeconomic balance. The model describes dynamic ofthe Russian economy as a result of interaction of 9 aggregated economicagents: Producer, Bank, Owner, Household, Trader, The State, The Cen-tral Bank, Importer and Exporter.

The model successfully reproduces dynamics of the main macroeco-nomic indicators (GDP and its components, inflation, loans, deposits,etc.) for ten years, including the period of the global crisis. The iden-tified model demonstrates the strong turnpike property: even the agentsare allowed to know the future, this information has little effect on theiroptimal behavior.

References

1. Andreyev M., Vrzheshch V., Pilnik N., Pospelov I., Khokhlov M. “Intertempo-ral general equilibrium model of Russian economy based on national accountsdesagregation” Journal of Mathematical Sciences. Springer, New York, 1, No.182. 41–143 (2012)

256

A Modified Steepest Descent Method Based onBFGS Method for Locally Lipschitz Functions

R. Yousefpour1

1 Department Mathematical Sciences, University of Mazandaran, Babolsar,

Iran; [email protected]

Abstract

In this paper, the steepest descent method is modified basedon the BFGS method. At first, the descent direction is approxi-mated based on the Goldstein subgradient and a positive definitematrix. Then the Wolfe condition is generalized for the locallyLipschitz functions. Next, an algorithm is developed to compute astep length satisfying the generalized Wolfe condition. At each iter-ation, the positive definite matrix is updated by the BFGS method.By the presented line search algorithm, a pair of subgradients arecomputed for updating the positive definite matrix such that thesecant equation is satisfied. Finally, an algorithm is devolved basedon the BFGS method. This algorithm is implemented with MAT-LAB codes.keywords: Lipschitz functions, Armijo and Wolfe conditions, Linesearch, BFGS method.

Introduction

The steepest descent direction for the locally Lipschitz functions is com-puted based on an element of the Goldstein subgradient with minimalnorm. By an approximation of this direction, several bundle methods weredeveloped [1-6]. The efficiency of an algorithm, that developed based onan approximation of steepest descent direction, depends on the approxi-mation accuracy. To improve the accuracy of an approximation, we needto compute a larger number of subgradient and, this is time consuming.For example, in [6], the steepest descent direction is approximated bysampling gradients. This approximation is appropriate, but computingthis approximation for large scale problems is very expensive. In [4], the

257

steepest descent direction is iteratively approximated. This method com-putes a good approximation for the steepest descent direction by the lessnumber of subgradients. The numerical results showed that this algorithmis more efficient than other methods.

If the steepest descent method is suitably approximated, then the con-structed algorithm based on this direction will be efficient. On the otherhand, the BFGS method has good behavior for some nonsmooth func-tions. It can be expected that the combination of the steepest descentme and BFGS methods will be efficient. The first step of combining thesteepest descent and BFGS methods is that compute an approximationof the steepest descent direction by the Goldstein subgradient and, a pos-itive definite matrix. In each iteration of the main algorithm, the positivedefinite matrix must be updated by the BFGS method thus, the Wolfestep length condition is generalized based on the Goldstein subgradient.Based on this generalization, we prove that there exist step lengths satis-fying this generalization for a descent direction. We generalized the linesearch algorithms [7] and, prove that these generalize algorithms computea step length satisfying the generalized Wolfe condition. These algorithmsalso compute two subdifferentials for updating the approximation of Hes-sian matrix.

By using the similar ideas to [4], the Goldstein subgradient is itera-tively approximated until a descent direction is found based on a positivedefinite matrix. By these ideas, the Goldstein subgradient is efficientlyapproximated. We prove that this procedure finds a descent directionafter finitely many iterations. This descent direction is an approximationof the steepest descent direction based on positive definite matrix. Thefunction is reduced along this direction. In each iteration of a minimiza-tion algorithm, the positive matrix is updated by the BFGS method. Thepresented line search algorithm finds two subgradients such that the pos-itive definite matrix can be updated and, the secant equation is satisfied.Finally, the algorithm is implemented with Matlab and, the results arecompared with other methods.

Generalized Armijo and Wolfe conditions

In this paper, we suppose that H is a positive definite matrix. Supposethat f : Rn → R is a locally Lipschitz function, v ∈ R

n and, g = −Hv. If

258

there exists α > 0 such that

f(x+ tg)− f(x) ≤ −t‖v‖H , ∀t ∈ (0, α),

then g is a descent direction. Now, the Armijo condition is given asfollows.

D e f i n i t i o n 1. Let f : Rn → R be a locally Lipschitz functionand, v ∈ R

n. Suppose that g = −Hv is a descent direction at x. A steplength α > 0 satisfies in the Generalized Armijo Condition (GAC), if

f(x+ αg)− f(x) ≤ −c1α‖v‖H ,where c1 ∈ (0, 1). Equivalently A(α) ≤ 0, where

A(α) := f(x+ αg)− f(x) + c1α‖v‖H .D e f i n i t i o n 2. Let f : Rn → R be a locally Lipschitz function,

v ∈ Rn and, g = −Hv be a descent direction at x. IfW (·) is not decreasing

on a neighborhood of α and, the GAC satisfies in α along direction g atx, then we say that the GWC satisfies at α along direction g at x.

Computing descent direction

Here, we discuss on finding a descent direction for a locally Lipschitz func-tion f based on ∂ǫf(x) and, a given positive definite matrix H . Considerthe following problem and, suppose that 0 6∈ ∂ǫf(x),

ξ0 = arg minξ∈∂ǫf(x)

‖ξ‖H . (1)

Let g = −Hξ0. We have f(x, g) ≤ −‖ξ0‖H < 0 and, by the Leburg MeanValue Theorem,

f(x+ tg)− f(x) ≤ −t‖ξ0‖H , ∀t ∈ (0, ǫ). (2)

Thus, g is a descent direction. But, solving this problem often is imprac-tical. So, ∂ǫf(x) is approximated by the convex hull of its some elements.More exactly, if

Wk = v1, v2, . . . , vk ⊂ ∂ǫf(x),then we consider convWk as an approximation of ∂ǫf(x). Therefore, wesolve the following problem, which is an approximation of (1),

v0 = arg minv∈ conv Wk

‖v‖H .

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Minimization Algorithm

In the previous Subsection, we found a descent direction based on a pos-itive definite matrix and, an approximation of ∂ǫf(x). To update theinverse of Hessian matrix approximation, we need a pair of subgradientssuch that the secant equation is satisfied. Thus, by applying the linesearch algorithm, a step length satisfying in the GWC and, a pair of sub-gradients for updating the inverse of Hessian matrix approximation arecomputed. Now, we present the nonsmooth version of BFGS algorithm.

Algorithm 1.

Step 0 (Initialization)Let x1 ∈ R

n, v11 ∈ ∂f(x1), θǫ, c1, θδ, ǫ1, δ1 ∈ (0, 1), H1 = In×n,c2 ∈ (c1, 1) and, set k = 1, where In×n is the identical matrix.

Step 1 (Set new parameters)Set m = 1, Hm

k = Hk and, xmk = xk.

Step 2 (Compute descent direction)Apply Algorithm 2 in [4] at point xmk , with H = Hm

k , v = vmk ,δ = δk and, ǫ = ǫk. Let nmk be the number of iterations needed fortermination of the algorithm and, let ‖wmk ‖Hm

k= min‖w‖Hk

: w ∈conv Wm

k . If ‖wmk ‖Hk= 0 then Stop else let gmk = −Hm

k wmk be

the descent direction.

Step 3 (Line search)If the stopping condition (2) is not satisfied then go to Step 5,else apply the line search algorithm. If the algorithm terminatessuccessfully, then α is the line search parameter satisfying the GWCand, vm+1

k ∈ ∂ǫf(xmk + αgmk ) is a vector such that < (vm+1k ), gmk >

+c2‖wmk ‖Hmk> 0, else α is the line search parameter satisfying the

GAC. Construct the next iterate xm+1k = xmk +αgmk and, go to Step

4.

Step 4 (BFGS update)If the line search algorithm terminates successfully, then set s =αgmk and, y = vm+1

k − vmk and

Hm+1k = Hm

k −Hmk yy

THmk

< y,Hmk y >

+ssT

< y, y >,

260

else set Hm+1k = I. Set m = m+ 1 and, go to Step 2.

Step 5 (Update parameters)Set ǫk+1 = ǫk × θǫ, δk+1 = δk × θδ, v1k+1 = vmk , xk+1 = xmk , Hk+1 =Hmk and, let k = k + 1. Go to Step 1.

Theorem 1. Let f : Rn → R be a locally Lipschitz function. If thelevel set M = x : f(x) ≤ f(x1), is bounded, then either Algorithm 1terminates finitely at some k0 and, m0 with wm0

k0 = 0 or every cluster

point of the sequence xk, generated by Algorithm 1, belongs to the setX = x ∈ R

n : 0 ∈ ∂f(x).

References

1. A. A. Goldstein. ”Optimization of Lipschitz continuous functions,” Mathe-matical Programming , 13:14–22, (1977).

2. D. P. Bertsekas and S. K. Mitter. A descent numerical method for opti-mization problems with nondifferentiable cost functionals,” SIAM Journal onControl, 11:637–652, (1973).

3. M. Gaudioso and M. F. Monaco. ” A bundle type approach to the uncon-strained minimization of convex nonsmooth functions,” Mathematical Pro-gramming, 23(2):216–226, (1982).

4. N. Mahdavi-Amiri and R. Yousefpour. ” An effective nonsmooth optimizationalgorithm for locally lipschitz functions,” Accepted Journal of OptimizationTheory Application.

5. P. Wolfe. ” A method of conjugate subgradients for minimizing non-differentiable functions,” Nondifferentiable Optimization, M. Balinski and P.Wolfe, eds., Mathematical Programming Study, North- Holland, Amsterdam,3:145–173, (1975).

6. J. V. Burke, A. S. Lewis, and M. L. Overton. ”A robust gradient samplingalgorithm for nonsmooth, nonconvex optimization,” SIAM Journal of Opti-mization, 15:571–779, (2005).

7. J. Nocedal and S. J. Wright Numerical optimization, Springer, (1999).

261

The primal affine-scaling method for semidefiniteprogramming with steepest descent

Vitaly Zhadan1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; [email protected]

In this paper we consider the variant of a feasible affine scaling primalmethod for solving linear semidefinite programming problem. This varianthas a local convergence at a linear rate, when all points in the iterativeprocess belong to the relative interior of the feasible set. But our aimis to show how it is possible to modify the method in order to have theopportunity to apply the steepest descent approach for choosing the stepsize.

Let Sn denote the space of symmetric matrices of order n, and let Sn+be the cone in Sn, consisting of positive semidefinite matrices. We usealso the inequality M 0 to indicate that a matrix M belongs to Sn+.The space Sn is finite-dimensional, and its dimension is equal to so-calledtriangular number k(n) = n(n + 1)/2. The inner product of matricesM1 and M2 of the same size is defined as the trace of the matrix MT

1 M2

and is denote by M1 •M2.Consider the linear semidefinite programming problem

min C •X, Ai •X = bi, 1 ≤ i ≤ m, X 0, (1)

where the matrices C, X and Ai (1 ≤ i ≤ m) belong to Sn. The dualproblem of (1) has the form

max bTu,

m∑

i=1

uiAi + V = C, V 0, (2)

where b = (b1, . . . , bm)T , V ∈ Sn. It is assumed that both problems(1) and (2) are solvable and the matrices Ai (1 ≤ i ≤ m) are linearlyindependent. We denote the feasible set of primal problem (1) by FP andits relative interior by F0

P .IfX∗ and V∗ are optimal solutions of problems (1) and (2), respectively,

then X∗•V∗ = 0 and the matricesX∗ and V∗ must commute. Hence, thereexists an orthogonal matrix Q such that

X∗ = QDiag(λ∗)QT , V∗ = QDiag(µ∗)Q

T ,

262

where λ∗ = [λ1∗, . . . , λn∗ ] and µ∗ = [µ1

∗, . . . , µn∗ ] are the eigenvalues of X∗

and V∗ respectively. The eigenvalues λi∗ and µi∗ satisfy the complementar-ity conditions λi∗µ

i∗ = 0, 1 ≤ i ≤ n. The strict complementarity condition

means that one of the values λi∗ or µi∗ is strictly positive.Since we assumed that both problems (1) and (2) have solutions, the

following system of equalities and inequalities

X • V = 0,Ai •X = bi, 1 ≤ i ≤ m,

V = C −∑mi=1 u

iAi,X 0, V 0,

(3)

is necessarily solvable.Denote by X ∗ V the symmetrized product of square matrices X and

V defined by the formula X ∗ V = (XV + V TXT )/2.Statement. For symmetric matrices X 0 and V 0, the equality

X ∗ V = 0nn is possible iff XV = V X = 0nn.Using the statement, we can replace the first equality in (3) by the

following one: X ∗ V = 0nn.Let the symbol vecX denote the direct sum of the columns of X ∈ Sn,

that is, the column vector of dimension n2 that consists of the columns ofX written one after another from top to bottom. For symmetric matrices,it is more convenient to deal with the column vector vechX of dimensionk(n). It also consists of the columns of X written one after another;however, these are not the entire columns but their parts beginning withthe diagonal entry. The operation vecsX is defined similarly. It differsfrom the preceding operation vechX only in that the off-diagonal entriesof X are multiplied by two before placing into vecsX .

Let also Ln and Dn are the elimination and duplicated matrices re-spectively [1]. The matrix Ln for an arbitrary square matrix X effectsthe transformation: LnvecX = vechX . By contrast, the matrix Dn actson an arbitrary symmetric matrix X so that DnvechX = vecX .

With the help of vectorization operators vech and vecs, the optimalityconditions (3) can be rewritten as

X⊗vechV = 0k(n),AvecsvechX = b,

vechV = vechC −ATvechu,X 0, V 0,

(4)

263

where X⊗ = LnX⊗Dn, and X⊗ = [X ⊗ In + In ⊗X ]/2 is the Kroneckersum of a matrix X , In is the identity matrix of order n. By Avech andAvecs we denote the m×n2 matrices with vechAi and vecsAi respectivelyas their rows, 1 ≤ i ≤ m.

Let us consider the primal affine scaling method for solving problem(1), based on some approach for solving the system (4). For its derivingfrom (4) we substitute the expression for vechV from the third equalityto the first one and multiply both sides of that equality by the matrixAvecs from the left. As a result, we obtain the following system of linearalgebraic equations with respect to the vector u:

Γ(X)u = AvecsX⊗vechC, (5)

where Γ(X) = AvecsX⊗ATvech.If the matrix Γ(X) is nonsingular, then solving system (5), we obtain

u(X) = Γ−1(X)AvecsX⊗vechC.

Denote V (u) = C −∑mi=1 u

iAi, V (X) = V (u(X)). After substitutionV (X) into the first equality from (4) we obtain the nonlinear system ofequations with respect to X :

X ∗ V (X) = 0nn.

Applying the fixed point method for solving this system, we have

Xk+1 = Xk − αkXk ∗ Vk, Vk = V (Xk), (6)

where X0 ∈ F0P , αk > 0.

LetRA be the subspace of Sn spanned by the matricesAi, (1 ≤ i ≤ m),and let R⊥

A be its orthogonal complement. Let also TX be the tangentspace of Sn+ at the point X . Following [2], we give the definition of anondegenerate points X and V .

D e f i n i t i o n. A feasible point X of the primal problem (1) isnondegenerate, if R⊥

A + TX = Sn . Similarly, a feasible point V of dualproblem (2) is nondegenerate, if RA + TV = Sn.

Lemma. Let X ∈ FP be a nondegenerate point. Then the matrixΓ(X) is nonsingular.

In what follows, we assume that the problem (1) is nondegenerate, i.e.all points from FP are nondegenerate.

264

Theorem [3]. Let the solutions X∗ and V∗ of primal and dual prob-lems (1), (2) are strictly complementary. Let also points X∗ and V∗ arenondegenerate. Then for αk sufficiently small the iterative process (6)locally converges to X∗ at a linear rate.

Denote ∆Xk = Xk ∗ V (Xk). The iterative process (6) possesses thefollowing properties:

Ai •∆Xk = 0, 1 ≤ i ≤ m; C •Xk+1 ≤ C •Xk.

Thus, in order to obtain the maximal decreasing of the value of the objec-tive function C •X , the step size αk should be taken as large as possibleprovided that Xk+1 0. It leads to application of steepest descent ap-proach for choosing the step size. But at boundary points Xk ∈ FP wehave ∆Xk ∈ TXk

. Hence, we must modify the right hand side in (8), whenXk is a boundary point of FP .

Assume that X is a boundary point of the feasible set FP , and assumealso that the rank of Xk is r < n. Then the matrix X can be representedin the form:

X = QDiag(λ1, . . . , λr, 0 . . . , 0

)QT ,

where Q is an orthogonal matrix and λi > 0, 1 ≤ i ≤ r. Let QB and QNbe the submatrices formed of the first r and the last n− r columns of thematrix Q. Let also λB = [λ1, . . . , λr].

At the non-optimal point X ∈ FP the matrix V (X) is not positivelysemidefinite. Therefore the matrix V Q(X) = QTV (X)Q is sign indefinite.

Suppose that its principal submatrix V QNN = QTNV (X)QN is also sign

indefinite. Without loss of generality we may regard V QNN as a diagonalmatrix. Then some of its diagonal elements are negative.

Let y be a (n − r)-dimensional vector consisting from nonnegativeelements with

∑n−ri=1 y

i = 1. Moreover let positive elements of y dispose

at places corresponding to negative diagonal elements of V QNN . Denote beY the diagonal matrix such that Y 2 = Diag(y).

We take some ε > 0 and instead of the direction ∆X = X ∗ V (X)consider the following one:

∆X = ∆XB + ε∆XN , (7)

where

∆XB = QBΛBQTB, ΛB = Diag(λB) ∗

(QTBV (u)QB

),

265

∆XN = QNΛNQTN , ΛN = Y 2 ∗ (EN−Q

TNV (u)QNEN−). (8)

In (7) the first direction ∆XB belongs to the minimal face of the setFP containing the point X . The second direction ∆XN belongs to theconjugate face of the minimal face. The matrix EN− in (8) is a diagonalmatrix with the indicator vector of y on its diagonal.

The dual vector u is taken in such a way that the following system oflinear equalities

Ai • (∆XB + ε∆XN ) = 0, i = 1, . . . ,m.

is fulfilled.It can be verified that the direction (7) leads to decreasing of the value

of the objective function C •X . Using the steepest descent for choosingαk, we obtain the updated point Xk+1. This point again belongs to theboundary of the feasible set FP . The behavior of the method is similarto behavior of the simplex method. Certainly, because of the fact thatthe cone Sn+ is non-polyhedral, infinitely many iterations are required forsolving (1).

The author was supported by the Programs of Fundamental Research of

Russian Academy of Sciences P-15, P-18, by the Russian Foundation for Basic

Research (project no.11-01-00786), and by the Leading Scientific Schools Grant

no.5264.2012.1

References

1. J.R. Magnus, N.Neudecker Matrix Differential Calculas with Applicationsin Statistics and Econometrics, John Willey & Sons, Chichester, New York,Toronto (1988).

2. F. Alizadeh, J.-P.F. Haeberly, M.L.Overton “Complementarity and nonde-generacy in semidefinite programming”. Mathematical Programming, SeriesB, 77, 129–162 (1997).

3. M.S. Babynin, V.G. Zhadan. “A Primal Interior Point Method for Lin-ear Semidefinite Programming Problem”. Computational Mathematics andMathematical Physics, 48, 1746-1767 (2008).

266

The point projections on linear manifold

Valery Zorkaltsev1

1 Melentiev Energy Systems Institute SB RAS, Irkutsk, Russia;

[email protected]

Many mathematical models can be reduced to the geometric problemof finding the least distance points of the linear manifold from the origin.The problems of estimating of the parameters of the linear regression,the problems of the regularization of solutions in the balance models,the problems of the search of pseudo-solutions in models with conflictingconditions are considered in the report as examples.

The given geometric problem appears as a component part of the somecomputational algorithms. It is discussed in the report as applied tosolving of the systems of nonlinear equations by the iterative linearizationmethod and conformably to the interior point algorithms for solving ofthe linear and nonlinear mathematical programming problems.

The main goal of this report is to present the recent results and resultspublished in [1], in the study of properties and correlations of the leastdistant from the origin points of the linear manifold under the differentdefinitions of closeness.

Let’s consider some concretizations of the given geometric problem.Let L be the linear manifold in Rn, set J(x) be set of numbers of nonzerocomponents of the vector x from Rn (carrier of the vector). The symbol⊂ denotes strict inclusion. The set of vectors Rn with all positive compo-nents we denote Rn++. The convex hull of set and closure of set X fromRn are denoted coX and clX .

1. Let’s introduce the set of vectors of the linear manifold with theminimal carriers

B = x ∈ L : ¬ ∃y ∈ L, J(y) ⊂ J(n).

It is known [1] that the amount of vectors in B is finite and no more thanCmn , where m is dimension of manifold L.

2. Lets denote the set of vectors of the linear manifold with Pareto-minimal absolute meanings of the components

Q = x ∈ L : ¬ ∃y ∈ L,∑|yi| <

∑|xi|, |yi| ≤ |xi|, i = 1, ..., n.

267

The convex hull B is minimal convex set, which contains Q (see [1]),

coQ = coB.

The set Q is bounded according to this relation. In addition to this wenote that the set Q can be nonconvex, but it is tie set and it is union ofthe finite amount of the polytopes.

3. Let’s introduce the Holder, octahedral and Euclidean projectionof a point on the linear manifold. We obtain for the Holder norm thefollowing expressions when the vector of a weighted coefficients h ∈ Rn++

and the degree coefficient p ∈ (1,∞) are given

y(p, h) = arg min∑

hi|yi|p : y ∈ L,

Y (1, h) = Arg min∑

hi|yi| : y ∈ L,

y(∞, h) = arg lex minmaxi hi|yi| : y ∈ L.The Holder projection is Euclidean projection under p = 2.

Let us imply that the Chebyshev projection y(∞, h) is determined asa result of solving of the following multicriteria problem of lexicographicoptimization (the symbol lex denotes this). We find the vector from Lwith minimal meanings of the maximal values hi|yi|. Moreover, we findthe relative interior point of solutions of this problem. We fix the meaningsof a components of the vector y under which the maximal meaning of thegiven values is obtained (it is true for all solutions of the given problem).On the next stage we solve the minimization problem of maximal meaningsof the values hi|yi| for the remaining variables. When we found a relativeinterior point of the optimal solutions of this problem, we fix variables yion their optimal level under which the optimal value of objective functionis achieved on the second stage problem. If the solution is not uniquethen we’ll repeat the process. We obtain the unique solution after thefinite stages of computation. This solution can be used as the Chebyshevprojection of the origin on the linear manifold L.

The sets of all octahedral, Chebyshev, Euclidean projections with thedifferent weighted coefficients and the set of the Holder projections aredenoted

P1 =⋃

h∈Rn++

Y (1, h),

268

P∞ =y(∞, h) : h ∈ Rn++

,

P2 =y(2, h) : h ∈ Rn++

,

P =y(p, h) : h ∈ Rn++, p ∈ (1, ∞)

.

The following relations was earlier proved

P2 = P, clP2 = Q.

Let’s announce the statements

Y (1, h) ∩B 6= ⊘, ∀ h ∈ Rn++,

P1 = Q, P∞ = P2.

The octahedral projections can have more than one meaning for thegiven vector of weighted coefficients h. Moreover, the vectors of the lin-ear manifold with the minimal carrier always exist among the octahedralprojections with the given vector of weighted coefficients according to thefirst adduced relations.

In particular, any Holder and Chebyshev projections of point on thelinear manifold can be received by the least-squares method thanks tochoice of the vector of weighted coefficients. Any octahedral, Euclideanprojections and any vector of the linear manifold with Pareto-minimalabsolute meanings of components (including any vector from B) can beobtained with any given accuracy according to the technique too. Thesestatements follow from the above relations.

References

1. V.I. Zorkaltsev. The least-squares method: geometric properties, alternativeapproaches, applications, Nauka, Novosibirsk (1995) (in russian).

269

The effect of the setup parameters on the evolution ofthe substance crystallization process

Vladimir Zubov1, Alla Albu2, Andrey Albu3

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; [email protected] Dorodnicyn Computing Centre of RAS, Moscow, Russia; [email protected] Dorodnicyn Computing Centre of RAS, Moscow, Russia; [email protected]

The process of solidification in metal casting is considered. The crys-tallization of molten metal is one of the important stages of foundry prac-tice. On that how the crystallization process proceeded, depends thequality of the obtained model.

Fig. 1 Fig. 2

Liquid metal is placed into the work cavity of a mould with a pre-scribed configuration (see Fig. 1). A special setup is used to crystallizethe metal (see Fig. 2). It consists of upper and lower parts. The upperpart consists of a furnace with a mold moving inside. The lower part is acooling bath consisting of a large tank filled with liquid aluminum whosetemperature is somewhat higher than the aluminum melting point. Themould is slowly moving among the furnace and begins to dip into liquidaluminum that has a relatively low temperature and thus is cooling themetal. At the same time the mould receives heat from the furnace andthis heat doesn’t allow the crystallization process to run too fast. Thecrystallization process is affected by different phenomena such as heatloss due to heat radiation, gain of energy by the mould due to expanse of

270

heat radiation from aluminum and furnace, heat exchange between liquidaluminum and the mould.

Different technological requirements can be imposed on the evolutionof the phase boundary to obtain a detail of the desired quality. In this workcertain ones are considered: the shape of the phase boundary should beclose to a plane and its law of motion should be close to a preset one. Oneof the consequences of satisfying these requirements would be absence of”bubbles” of liquid metal. These ”bubbles” are essentially areas of liquidmetal that are surrounded by already solidified metal. If such areas arepresent during the solidification process, the quality of the detail wouldbe unacceptable for sure.

The process is described by a three-dimensional unsteady heat equa-tion (see [1]). The complication of the problem is that the metal can bepresent simultaneously in two phases: solid and liquid. Density of ma-terials, their heat capacity and thermal conductivity all depend on thetemperature and are discontinuous in the border between the metal andthe mould.

The evolution of the solidification front is affected by numerous pa-rameters (for example, by the furnace temperature, the liquid aluminumtemperature, the depth to which the object is immersed in the liquid alu-minum, the velocity of the mold relative to the furnace, etc.) that couldbe controlled, but in practice most often the speed u(t) at which the moldmoves relative to the furnace is used as the control. To find a controlu(t) that will satisfy the mentioned technological requirements, we statethe optimal control problem (see [2]) and write down the following costfunctional:

I(u)=1

t2(u)− t1(u)

t2(u)∫

t1(u)

∫∫

S

[Zpl(x, y, t, u)−z∗(t)]2dxdydt.

Here t1 is the time, when the crystallization front arises; t2 is the time,when the crystallization of metal completes; (x, y, Zpl(x, y, t, u)) are thereal coordinates of the interface at the time t; (x, y, z∗(t)) are the desiredcoordinates of the interface at the time t; S is the largest cross section ofthe mold that is filled with metal.

In this paper we investigate the effect of the furnace temperature andthe maximum depth of immersion of the mold into the cooler on the evolu-tion of the substance crystallization process. A large series of calculations

271

of the substance crystallization problem was carried out in which these pa-rameters were varied within wide limits. As a starting point in this serieswas selected a point with the following parameters: maximum allowabledepth of immersion into the cooler DISPL = 360 mm (immersion up tothe fourth stage), and the furnace temperature Tsou = 1840 oK (here andbelow the immersion is measured in millimetres, the furnace temperatureis measured in Kelvin degrees and the speed of moving object is mea-sured in mm/min). These are the actual values of the parameters of thesubstance crystallization process that are used in the real setup shown inFig. 2. The aim of the research was to answer the following question:is it possible by changing the oven temperature Tsou and the maximumdepth of immersion DISPL to achieve a more preferable crystallizationprocess?

The study was conducted for the case when the velocity of the objectrelative to the furnace was constant. The value of this velocity is variedin the range from 2 to 50 (mm/min). The results of the numerical stud-ies presented below are divided into four groups. Each group includesthe results of calculations which were obtained at the same value of themaximum depth of immersion DISPL. Various versions in each groupdiffer by temperature Tsou and velocity u(t) of the object relative to thefurnace.

DISPL = 360 (immersion up to the 4th stage)

The calculation results show that when using this immersion depth themetal solidification process is admissible (technological requirements areviolated within acceptable limits), if the temperature of the furnace is inthe range 1830 < Tsou < 1850 and the constant speed of movement is inthe range 18 < u(t) < 25. If speed u(t) is out of this range then it leads toformation of liquid metal ”bubbles” during the crystallization process forall considered values of the furnace temperature Tsou. A similar situationarises if the furnace temperature Tsou < 1830 for all considered values ofvelocity. If the temperature Tsou > 1850, the ”bubbles” of liquid metaldo not form, but at the same time either crystallization process doesn’tcome to an end or it takes an unacceptably long time. Note also that adecrease in speed and increase in the oven temperature (in the specifiedrange) leads to a crystallization process that is closer to the desired one.

272


Decrease in the maximum depth of immersion of the object into thecooler, as shown by calculations, leads to a deterioration of the character-istics of the crystallization process. Neither change in the velocity of themold u(t), nor change in the oven temperature Tsou allowed to get rid ofliquid ”bubbles” in the metal. It should also be noted that in this case thecrystallization of the metal runs unacceptably long (or isn’t happening atall) for even lower temperatures starting from Tsou > 1810.


The increase in the maximum depth of immersion has a positive effecton the characteristics of the crystallization process. The ranges of variableparameters (u(t),Tsou), for which the crystallization process is admissible,are as follows: (10 < u(t) < 25, 1900 < Tsou < 1950). In this case evensmaller speeds u(t) and higher temperatures Tsou are acceptable thanwhen DISPL = 360. For temperatures Tsou < 1900, as in the case ofDISPL = 360, there appear liquid metal ”bubbles”, and for Tsou > 1950the process of solidification is unacceptably long. It should be noted thatif the object is immersed up to the fifth stage (DISPL = 420), thenthe best results (in terms of the objective function) occur at the lowestallowable speed u(t) = 10. In this case, as before, the value of the costfunctional decreases when the furnace temperature Tsou increases.

DISPL = 450 (almost complete immersion)

With this depth of immersion the following ranges of parameters wereacceptable: (5 < u(t) < 25, 1970 < Tsou < 2015). Qualitatively, thecharacteristics of the crystallization process behave similarly to the casewhen DISPL = 420. The objective functional I(u) decreases with theincrease of the furnace temperature and its minimum value is achievedwhen the object’s velocity u(t) ≈ 10. We’d like to note that this minimumis the smallest among all obtained in the series of calculations with varyingdepth DISPL.

The results of calculations performed for different depths of immer-sion DISPL allow us to conclude that in each case there is a range ofacceptable furnace temperatures Tsou, the limits of which increase withdepth DISPL. The minimum is delivered to cost functional I(u) at theupper boundary of this range. There is also a range for the speed u(t)

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of the object in the oven. This range gets wider with an increase in themaximum depth of immersion. The value of the functional I(u) reachesits minimum value at the lower boundary of this range for small values ofDISPL and gets inside this range for higher ones. The value of u(t) atwhich I(u) is minimal decreases when DISPL increases.

This work was supported by the Russian Foundation for Basic Research

(project No. 12-01-00572- and No. 11-01-12136-ofi-m-2011), by the Program for

Fundamental Research of Presidium of RAS P18, and by the Program ”Leading

Scientific Schools” (NSh-5264.2012.1).

References

1. Albu A.F., Zubov V. I., “Mathematical Modeling and Study of the Processof Solidification in Metal Casting,” Computational Mathematics and Mathe-matical Physics, 47, No. 5, 843–862 (2007).

2. Albu A.V., Zubov V. I., “Choosing a Cost Functional and a Difference Schemein the Optimal Control of Metal Solidification,” Computational Mathematicsand Mathematical Physics, 51, No. 1, 21–34 (2011).

Convergence of the two-step extragradient method ina finite number of iterations

Anna Zykina1, Nikolay Melenchuk2

1 Omsk State Technical University, Omsk, Russian; [email protected] Omsk State Technical University, Omsk, Russian; [email protected]

The extragradient methods [1] are an effective tool of solving of thevariational inequalities. For convergence they require fewer conditionsthan the gradient methods need. The monotone convergence in the normto one of the solutions of the variational inequalities is known for the ex-tragradient methods. In the present paper the two-step method of solvingof the variational inequalities is considered. Convergence of this method

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in a finite number of the iterations is shown if the severity condition isfulfilled.

To solve the variational inequality means to find a vector z∗ ∈ Ω thatsatisfies the following conditions:

〈H(z∗), z − z∗〉 ≥ 0, ∀ z ∈ Ω. (1)

Here H : Rn → Rn, Ω – a convex closed set Ω ⊂ R

n.The expressions

zk = PΩ(zk − αH(zk)),

zk = PΩ(zk − αH(zk)), (2)

zk+1 = PΩ(zk − αH(zk)).

are the recurrence relations for the two-step extragradient method of solv-ing of the variational inequalities [2].

A condition providing convergence of a computational scheme (2) ina finite number of iterations is the severity condition, which holds (isfilfilled) for the solutions z∗ of the variational inequality with some γ > 0:

〈H(z), z − z∗〉 ≥ γ‖z − z∗‖, ∀ z ∈ Ω. (3)

We demonstrate that the condition (3) means an uniqueness of thesolution of the variational inequality. We consider two solutions of thevariational inequality z∗1 and z∗2 . Let they are different, and the severitycondition (3) is fulfilled for each solution:

〈H(z), z − z∗1〉 ≥ γ‖z − z∗1‖, ∀ z ∈ Ω, (4)

〈H(z), z − z∗2〉 ≥ γ‖z − z∗2‖, ∀ z ∈ Ω.

The point z∗2 belongs to the domain Ω of finding of a solution, then itcan be substituted in the inequality (3) instead of z. We obtain

〈H(z∗2), z∗2 − z∗1〉 ≥ γ‖z∗2 − z∗1‖, (5)

From the inequality (1) at z∗ = z∗2 and z = z∗1 we have

〈H(z∗2), z∗1 − z∗2〉 ≥ 0. (6)

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The inequalities (5) and (6) are fifilled simultaneously at z∗1 = z∗2 . Con-sequently, the uniqueness of solution of the variational inequality (1) holdsif the condition of severity (3) is fulfilled. This fact narrows significantlya class of problems satisfying these conditions.

Under the condition of severity (3) for the variational inequality (1)the theorem is proved.

Theorem. Let:a) Ω is a closed convex set;b) H(z) is a monotone operator: 〈H(z)−H(v), z − v〉 ≥ 0, ∀ z, v ∈ Ω,satisfying the Lipshitz condition with a constant L > 0:

‖H(z)−H(v)‖ ≤ L‖z − v‖, ∀ z, v ∈ Ω;

c) there exists a solution z∗ of the variational inequality (1);d) the severity condition (3) with a constant γ > 0 is filfilled;e) 0 < α < 1√

3L.

Then the sequence zk, defined by the recurrence relations (2), convergesto the solution z∗ ∈ Ω of the variational inequality (1) in a finite numberof iterations.

As an example a question of convergence in a finite step number ofthe two-step extragradient method for a linear programming problem isconsidered [3]:

xk = [xk + α(Ayk − c)]+,yk = [yk − α(ATxk − b)]+,xk = [xk + α(Ayk − c)]+,yk = [yk − α(AT xk − b)]+, (7)

xk+1 = [xk + α(Ayk − c)]+,yk+1 = [yk − α(AT xk − b)]+.

Here a pair of the dual linear programming problems is given as follows:

〈c, x〉 → min, 〈b, y〉 → max,

ATx ≥ b, Ay ≤ c, (8)

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x ≥ 0, y ≥ 0,

[p]+ = max0, p is the projection on the positive semiaxis for the scalarquantity p and [p]+ = ([p1]

+, [p2]+, ..., [pl]

+) is the projection on the pos-itive orthant for the vector p = (p1, p2, ..., pl).

The linear programming problem (8) is a special case of a variationalinequality. Interest to this problem is explained by a possibility to demon-strate the results in the practice. In the case of an one-step extragradientmethod the question of convergence in a finite step number for a linearprogramming problem (8) has been studied by A.S. Antipin in [4].


(project no. 12-07-00326).

References

1. G.M. Korpelevich. “The extragradient methof of finding of suddle points andof other,” Economics and the mathematical methods. Vol. 12, No. 4, pp. 747–756 (1976).

2. A.V. Zykina, N.V. Melenchuk. “The two-step extragradient method for thevariational inequalities,” Izvestia VUZov. News of the High Mathematics.No. 9, pp. 82–85 (2010).

3. A.V. Zykina, N.V. Melenchuk. “The doublestep extragradient method forsolving a problem of the management of resource,” Automatic Control andComputer Sciences. Vol. 45, No. 7, pp. 1–8 (2011).

4. A.S. Antipin. Gradient and extragradient approaches in the bilinear equlib-rium programming, Computing center RAS, Moscow (2002).

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Author index

Abdullayev Vagif . . . . . . . . . . . . . . 10Abramov Alexander P. . . . . . . . . 13Abramov Oleg V. . . . . . . . . . . . . . . 16Afanasyev Alexander . . . . . . . . . . 21Ahookhosh Masoud . . . . . . . . . . . . 30Aida-zade Kamil . . . . . . . . . . . . . . 21Akbari Z. . . . . . . . . . . . . . . . . . . . . . . 25Albu Alla . . . . . . . . . . . . . . . . . . . . 270Albu Andrey . . . . . . . . . . . . . . . . . 270Amini Keyvan . . . . . . . . . . . . . . . . . 30An Phan Thanh . . . . . . . . . . 93, 106Anikin Anton . . . . . . . . . . . . . . . . . . 31Anop Maxim . . . . . . . . . . . . . . . . . . 34Antipin Anatoly . . . . . . . . . . . . . . . 36Arkhipov Dmitry . . . . . . . . . . . . . . 42Ashrafova Yegana . . . . . . . . . . . . . 21Attari M. Ahmadian . . . . . . . . . 218

Bahrami Somayeh . . . . . . . . . . . . . 30Beklaryan Armen . . . . . . . . . . . . . . 47Bushenkov Vladimir . . . . . . . . . . . 51

Caldeira Bento . . . . . . . . . . . . . . . . 51Constantino Miguel . . . . . . . . . . . . 54

Dikusar V.V. . . . . . . . . . . . . . . . . . . 59Dolgui Alexandre . . . . . . . . . . . . . . 81Dorjieva Anna . . . . . . . . . . . . . . . . . 63Druzhinina Olga . . . . . . . . . . . . . . . 67Dunin-Barkowski W. . . . . . . . . . . 72

Evtushenko Yu.G. . . . . . . . . . . . . . 76

Freitas Adelaide . . . . . . . . . . . . . . 164Førsund Finn . . . . . . . . . . . . . . . . 144

Gafarov Evgeny R. . . . . . . . . . . . . 81

Galiev Shamil . . . . . . . . . . . . . . . . . 86Gasnikov Alexander . . . . . . . . . . . 91Gasnikova Eugenia . . . . . . . . . . . . 91Gholami Omid . . . . . . . . . . . . . . . 245Giang Dinh Thanh . . . . . . . . . . . . 93Golikov Alexander I. . . . . . . . . . . . 94Golshtejn Evgenij . . . . . . . . . . . . . 99Goncharov Vladimir . . . . . . . . . . 101Gornov Aleksander . . . . . . . . . . . 103Guerman Anna D. . . . . . . . . . . . . 106

Hai Nguyen Ngoc . . . . . . . . . . . . 106Husnullin Nail . . . . . . . . . . . . . . . . 187

Ismagilov Niyaz . . . . . . . . . . . . . . 111

Jacimovic M. . . . . . . . . . . . . . . . . . 115

Kamenev Georgij . . . . . . . . . . . . . 159Kaneva Olga . . . . . . . . . . . . . . . . . 147Kasenov Syrym . . . . . . . . . . . . . . 192Katueva Yaroslava . . . . . . . . . . . . . 34Khachaturov Ruben V. . . . . . . . 118Khachaturov Vladimir R. . . . . . 122Khachay Michael . . . . . . . . . . . . . 124Khoroshilova Elena . . . . . . . . . . . 128Klimentova Xenia . . . . . . . . . . . . . 54Kobylkin Konstantin . . . . . . . . . 133Korenev Pavel . . . . . . . . . . . . . . . . 134Kostyukova Olga . . . . . . . . . . . . . 139Krivonozhko Vladimir . . . . . . . . 144Krnic I. . . . . . . . . . . . . . . . . . . . . . . 115Kulbida Ulyana . . . . . . . . . . . . . . 147Kuliev Samir . . . . . . . . . . . . . . . . . 153

Laskova Maya . . . . . . . . . . . . . . . . 156

278

Lazarev Alexander . . . 42, 81, 134,156, 187

Lisafina Maria . . . . . . . . . . . . . . . . . 86Lotov Alexander V. . . . . . . . . . . 159Lukovenko A.A. . . . . . . . . . . . . . . 162Lychev Andrey . . . . . . . . . . . . . . . 144

Macedo Eloisa . . . . . . . . . . . . . . . . 164Magafurzyanova Nelli . . . . . . . . . 223Melenchuk Nikolay . . . . . . . . . . . 274Mikhailov Igor E. . . . . . . . . . . . . . 170Mizhidon Arsalan . . . . . . . . . . . . 175Mordukhovich Boris . . . . . . . . . . 178Murashkin Evgenii . . . . . . . . . . . 178Muravey L.A. . . . . . . . . . . . 170, 181Musatova Elena . . . . . . 42, 156, 187

Nasyrov Farit . . . . . . . . . . . . . . . . 111Nurseitov Daniar . . . . . . . . . . . . . 192

Obrosova Nataliya . . . . . . . . . . . . 194Orlov Andrei . . . . . . . . . . . . . . . . . 198

Parkhomenko Valeriy . . . . . . . . . 203Pereira Fatima . . . . . . . . . . . . . . . 101Perzhabinsky Sergey . . . . . . . . . . 207Petrov Lev F. . . . . . . . . . . . . . . . . 213Petrov V.M. . . . . . . . . . . . . . . . . . . 181Petrova Natalia . . . . . . . . . . . . . . . . 67Peyghami M. R. . . . . . . . . . . 25, 218Pilnik N.P. . . . . . . . . . . . . . . . . . . . 256Pinigin Sergei . . . . . . . . . . . . . . . . 198Plakhov Alexander . . . . . . . . . . . 219Poberii Maria . . . . . . . . . . . . . . . . 124Polyak Boris . . . . . . . . . . . . . . . . . 220Pospelov I.G. . . . . . . . . . . . . . . . . . 256Posypkin Mikhail . . . . . . . . . . . . . 221Putilina Elena . . . . . . . . . . . . . . . . . 21

Rassadnikova Ekaterina . . . . . . 223Romanenkov A.M. . . . . . . . . . . . 181

Sedova Nataliya . . . . . . . . . . . . . . 227Serovajsky Simon . . . . . . . . . . . . . 229Shakenov Iliyas . . . . . . . . . . . . . . . 234Shakenov Kanat . . . . . . . . . . . . . . 237Shananin Alexander . . . . . . . . . . 194Shcherbina O. . . . . . . . . . . . . . . . . 240Shishlenin Maksim . . . . . . . . . . . 192Sigal Izrael . . . . . . . . . . . . . . . . . . . 221Smirnov Georgi . . . . . . . . . . . . . . . . 51Sotskov Yuri N. . . . . . . . . . . . . . . 245Sviridenko A. . . . . . . . . . . . . . . . . . 240

Tavakoli H. . . . . . . . . . . . . . . . . . . . 218Tchemisova Tatiana . . . . . . . . . . 139Tikhomirova T.M. . . . . . . . . . . . . 162Trang Le Hong . . . . . . . . . . . . . . . . 93Tretyakov A.A. . . . . . . . . . . . 76, 250

Valeeva Aida . . . . . . . . . . . . . . . . . 223Viana Ana . . . . . . . . . . . . . . . . . . . . 54Vorontsova Evgeniya A. . . . . . . 254Vrzheshch V.P. . . . . . . . . . . . . . . . 256Vyshinskiy L. . . . . . . . . . . . . . . . . . . 72

Werner Frank . . . . . . . . . . . . . . . . 245

Yousefpour R. . . . . . . . . . . . . 25, 257Yudin Vitalii . . . . . . . . . . . . . . . . . . 86

Zasukhina E.S. . . . . . . . . . . . . . . . . 59Zhadan Vitaly . . . . . . . . . . . . . . . . 262Zorkaltsev Valery . . . . . . . . 207, 267Zubov Vladimir . . . . . . . . . . . . . . 270Zykina Anna . . . . . . . . . . . . . . . . . 274

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Documents

OPTIMIZATION AND APPLICATIONS · Vagif Abdullayev1 1 Cybernetics Institute of ANAS, Baku, Azerbaijan; [email protected] Let us consider the following optimal control problem for