Organizing Committee - · PDF fileKarine Abgaryan, Olga Volodina Approaches to the solution of the ... Vladimir Zubov, Alla Albu Control of Phase Boundary Evolution in

Montenegrin A ademy of S ien es and Arts

University of Montenegro

University of Evora, Portugal

Dorodni yn Computing Centre, FRC CSC RAS

Mos ow Institute of Physi s and Te hnology

(State University)

VII International Conferen e on Optimization

Methods and Appli ations

OPTIMIZATION

AND

APPLICATIONS

(OPTIMA-2016)

Petrova , Montenegro, September 2016

PROCEEDINGS

Mos ow 2016

UDC 519.658

Pro eedings in lude abstra ts of reports presented at the VII International

Conferen e on Optimization Methods and Appli ations Optimization and

appli ations (OPTIMA-2016) held in Petrova , Montenegro,

September 25 - O tober 2, 2016.

Edited by V.U. Malkova.

Íàó÷íîå èçäàíèå

© Dorodni yn Computing Centre of FRC Computer S ien e and

Control of Russian A ademy of S ien e, 2016

Organizing Committee

Jacimovic Milojica, Chair, Montenegrin Academy of Sciences and Arts(Montenegro)Evtushenko Yuri G., Chair, Dorodnicyn Computing Centre,FRC CSC RAS (Russia)Bushenkov Vladimir, Chair, University of Evora (Portugal)

Afanas’ev Alexander P., IITP RAS (Russia)Antipin Anatoly S., CC RAS FRC CSC RAS (Russia)Burova Natalia K., CC RAS FRC CSC RAS (Russia)Dragovic Vesna, Montenegrin Academy of Sciences and ArtsGaranzha Vladimir A., CC RAS FRC CSC RAS (Russia)Golikov Alexander I., CC RAS FRC CSC RAS (Russia)Gornov Alexander Yu., ISDCT SB RAS (Russia)Jacimovic Vladimir, University of MontenegroKelmanov Alexander, IM SB RAS (Russia)Khachay Mikhail, IMM UB RAS (Russia)Konatar Nikola, University of MontenegroKovacevic-Vujcic Vera, University of Belgrade (Serbia)Krnic Izedin, University of MontenegroLotov Alexander V., FRC CSC RAS (Russia)Mijajlovic Nevena, University of MontenegroMolchanov Evgeny G., MIPT (SU) (Russia)Nedich Angelia, University of Illinois at Urbana Champaign (USA)Nesterov Yuri, CORE Universite Catholique de Louvain (Belgium)Obradovic Oleg, University of MontenegroPospelov Igor G., CC RAS FRC CSC RAS (Russia)Posypkin Mikhail A., CC RAS FRC CSC RAS (Russia)Shananin Alexander A., MIPT (SU) (Russia)Smirnov Gueorgui, University of Minho (Portugal)Tchemisova Tatiana, University of Aveiro (Portugal)Tret’yakov Alexey A., CC RAS FRC CSC RAS (Russia)Zakharov Victor N., FRC CSC RAS (Russia)Zhadan Vitaly G., CC RAS FRC CSC RAS (Russia)Zonn Ivetta A., CC RAS FRC CSC RAS (Russia)Zubov Vladimir I., CC RAS FRC CSC RAS (Russia)

Contents

Karine Abgaryan, Sergey Uvarov Theoretical investigation of atomic

nitrogen adsorption on Si (111) surface in the framework of molecular dy-

namics approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Karine Abgaryan, Olga Volodina Approaches to the solution of the

optimization problem of interatomic potential fitting . . . . . . . . . . . . . . . . . . . 12

Alexander P. Abramov On the Cyclicity in Controllable Systems . . . . 14

Oleg Abramov Preventive maintenance based on parameters estimation

and prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A.P. Afanas’ev, S.M. Dzyuba, I.I. Emelyanova, E.V. Putilina

Construction of the minimal sets of differential equations with polynomial

right-hand side in MathCloud system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

A.P. Afanas’ev, S.M. Dzyuba, I.I. Emelyanova, E.V. Putilina

Horner’s generic scheme as applied to multidimentional polynomials in

MathCloud system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Kamil Aida-zade, Samir Guliyev Zonal Feedback Control for a Heating

Problem with Delay in Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Anatoly Antipin On methods of minimizing a sensitivity function under

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

A.Zh. Atamuratov, I.E. Mikhailov, L.A. Muravey Damping of os-

cillations of a rectangular membrane by using multiple point dampers . . . 24

Artem Baklanov, Pavel Chentsov, Alexander Gornov, Tatiana

Zarodnyuk A benchmark of heuristic algorithms for the double integrator

traveling salesman problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Vadim E. Berezkin, Elizaveta A. Lotova Experiments with Hybrid

Methods of Edgeworth-Pareto Hull Approximation in Nonlinear Multiob-

jective Problems with Many Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Nikita Bykov, Nataly Vlasova Parallel implementation of genetic al-

gorithm to search for ballistic installations optimal parameters . . . . . . . . . . 29

Daria Chernetsova Application of Random forest method to estimate

the incurred but not reported claims reserve of an insurance company . . . 30

4

Alexey Chernov, Pavel Dvurechensky, Alexander Gasnikov Accel-

erated Primal-Dual Gradient Method for Linearly Constrained Minimiza-

tion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Greta Ts. Chikrii Time dilatation principle in evolutionary games of

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Dmitry Denisov On the choice of the best forms of reinsurance . . . . . . . 35

Dmitry Denisov, Vladislav Latiy The optimal marketing strategy of

the firm of a special type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Galina Digo, Natalya Digo The complexities of predicting the individ-

ual processes of a technical object parameters changing at the analysis of

its condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Vasily Dikusar, Marek Wojtowicz, Elena Zasukhina Optimal con-

trol problem with state constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Olga Druzhinina, Natalia Petrova Development of automated scien-

tific information system taking into account query optimization . . . . . . . . . 42

Vladimir Elkin Separation of Trivial Parts from Control Systems . . . . . 44

Anton Eremeev, Alexander Kel’manov, Artem Pyatkin NP-

hardness of Minimum Length of Vectors Sum Problems . . . . . . . . . . . . . . . . 46

Adil Erzin Cost-effective covering of the strip with identical directional

sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Yuri G. Evtushenko, Alexander I. Golikov New approach to the

theorems of alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Evgeniya Finkelstein, Enkhbat Rentsen, Anton Anikin Technology

for a set of solutions search in one global optimization problem . . . . . . . . . 50

Evgeniya Finkelstein, Mikhail Svinin, Alexander Gornov Numer-

ical study of the problem of spherical mobile robot optimal control . . . . . . 51

Alexander Galashov, Alexander Kel’manovAn exact pseudopolynomial-

time algorithm for a NP-hard problem of searching a family of disjoint

subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B.V. Ganin Modified Newton’s method to solve a transportation problem 55

V.A. Garanzha, L.N. Kudryavtseva Adaptive method for simultane-

ous untangling and optimization of 3d meshes . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5

Franco Giannessi Images and Existence of Constrained Scalar and Vector

Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Edward Gimadi, Alexey Istomin, Oxana Yu. Tsidulko On the

m-Peripatetic Salesman Problem on random inputs . . . . . . . . . . . . . . . . . . . . . 60

Evgeny Golshteyn, Ustav Malkov, Nikolay Sokolov On Nash Point

Search Algorithms for Three-Person Games (3PG) . . . . . . . . . . . . . . . . . . . . . 62

Evgenii Goncharov Approximation algorithms for the resource-constrained

project scheduling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Vasily Goncharov, Leonid Muravey, Elchin Eyniev On Some Shape

Optimization Problems for Thin Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Alexander Gornov, Alexander Tyatyushkin, Tatiana Zarodnyuk,

Anton Anikin, Evgeniya Finkelstein Applied optimization problems:

How to treat them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Tatiana Gruzdeva, Alexander Strekalovsky An Approach to Frac-

tional Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Vladimir Jacimovic, Nikola Konatar Determination of optimal forcing

directions for synchronization of nonlinear oscillations . . . . . . . . . . . . . . . . . . 71

Milojica Jacimovic, Nevena Mijajlovic, Muhammad Aslam Noor

Some Generalizations of Gradient-type Projection Method for Solving

Quasi Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Igor Kaporin Lower Bound on Restricted Isometry Constants for Tight

Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Dmitry Karamzin, Fernando Pereira Generalized solutions of optimal

control problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Alexander Kel’manov On some clustering problems: NP-hardness and

efficient algorithms with performance guarantees . . . . . . . . . . . . . . . . . . . . . . . 78

Alexander Kel’manov, Sergey Khamidullin, Vladimir Khandeev,

Ludmila MikhailovaAn approximation algorithm for one NP-hard prob-

lem of partitioning a sequence into clusters with restrictions on their car-

dinalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Alexander Kel’manov, Sergey Khamidullin, Vladimir Khandeev,

Ludmila Mikhailova An approximation algorithm for a problem of par-

titioning a sequence into clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6

Alexander Kel’manov, Anna Motkova An approximation scheme for

a balanced 2-clustering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Alexander Kel’manov, Semyon Romanchenko, Sergey Khamidullin

An approximation scheme for a problem of finding a subsequence . . . . . . . 84

Ruben V. Khachaturov An algorithm of using the set of equivalence

method for solving the multicriterial optimization problems . . . . . . . . . . . . 85

Michael Khachay, Roman Dubinin Approximability of the Euclidean

Capacitated Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Michael Khachay, Katherine Neznakhina Approximation Shemes for

the Generalized TSP in Grid Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Michael Khachay, Maria Poberiy Complexity and Approximability of

Geometrical Piercing Set Problem for Rectangles Intersecting a Diagonal

Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Elena Khoroshilova Terminal control problem with fixed ends in dynam-

ics: saddle-point technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Vladimir Krivonozhko, Finn Førsund, Andrey Lychev Smoothing

factor of the frontier transformation in the DEA models . . . . . . . . . . . . . . . . 92

Alexander Lazarev, Nail Khusnullin, Elena Musatova, Aleksey

Petrov, Aleksey Gerasimov, Maxim Kharlamov, Denis Yadrent-

sev, Konstantin Ponomarev, Sergey Bronnikov Cosmonauts Train-

ing Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Alexander Lazarev, Elena Musatova, Ilia Tarasov, Yakov Zinder

Dynamic programming approaches for single-track scheduling problem . . 95

Valery Lebedev, Konstantin Lebedev Application of splines for the

evaluation of investment rate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Konstantin Lebedev, Tatyana Tyupikova Algorithm of decision of

task of smoothing out dynamic rows by functions with piece-permanent

parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Alexey Leschov, Leonid Minchenko Relation between Mangasarian-

Fromovitz condition and some other constraint qualifications in nonlinear

programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Alexander V. Lotov, Andrey I. Ryabikov Pareto Frontier Visualiza-

tion in Developing the Control Rules for Angara River Basin . . . . . . . . . . . 101

7

Vlasta Malkova Multi-criteria approach to the analysis of the efficiency

of optimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Olga Masina, Olga Druzhinina On optimal control of dynamical sys-

tems described by differential inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Marin Misur, Darko Mitrovic, Andrej Novak On a gradient con-

straint problem for scalar conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Nataliia Obrosova, Alexander Shananin The estimation of the com-

pany’s market capitalization based on production models taking into ac-

count the deficit of current assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Nicholas Olenev Identification of a dynamic model of Russian economy

with two kinds of capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Valeriy Parkhomenko Improved computing realization of atmospheric

general circulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Alexander Pesterev Ellipsoidal estimation of the attraction domain for

affine systems with constrained control resource . . . . . . . . . . . . . . . . . . . . . . . . 113

Lev Petrov Synchronization, self-organization and self-optimization in

nonlinear dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Aleksandr Petrovykh ARIMA-GARCH models of RTS and MICEX fu-

tures dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Leonid D. Popov, Vladimir D. Skarin On alternative duality and

symmetric lexicographical correction of improper linear programs . . . . . . 117

Agnieszka Prusinska, Alexey Tret’yakov Singular optimization and

p-regularity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

L.A. Rybak, Y.A. Mamaev, D.I. Malyshev Optimization actuators

movements of robotic system with perturbation effect . . . . . . . . . . . . . . . . . . 121

Maxim Sakharov, Anatoly Karpenko New Parallel Multi-Memetic

MEC-based Algorithm for Loosely Coupled Systems . . . . . . . . . . . . . . . . . . . . 124

V.P. Savelyev, A.A. Shamin A Continuous Model of Rhythmical Pro-

duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

V.I. Shmyrev Iterative Equilibrium Searching in Piecewise Linear Ex-

change Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8

Alexander Skiba The solution of an applied problem with mixed con-

straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Alexander Strekalovsky Exact penalization and global optimality con-

ditions in nonconvex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Nikolay Tikhomirov Protection and safety for people optimization in

emergency situations with radiation leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Tatiana Tikhomirova, Valeriya Gordeeva Healthcare expenditure op-

timization subject to health burden in Russian regions . . . . . . . . . . . . . . . . . 136

Oxana Yu. Tsidulko Algorithms with performance guarantees for some

hard assignment and rooting problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Andrei Turkin Automatic Differentiation in Python . . . . . . . . . . . . . . . . . . 140

Alexander Vasin Optimization of transmission system for energy market 142

Tatiana Zarodnyuk, Armen Beklaryan, Fedor Belousov The com-

putational technique for approximation of nonlinear functional differential

equations of pointwise type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Tatiana Zarodnyuk, Alexander Bugerya, Fedor Khandarov Algo-

rithms for global optimization based on Curvilinear Search . . . . . . . . . . . . . 146

Elena Zasukhina, Sergey Zasukhin Determining parameters of hydro-

logical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Vitaly Zhadan On a variant of dual simplex-like algorithm for linear

semi-definite programming problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Vladimir Zubov, Alla Albu Control of Phase Boundary Evolution in

Metal Solidification for New Thermodynamic Parameters of the Metal . . 151

Daniil Musatov, Alexei Savvateev, Anton Trubakov, Shlomo We-

ber Endogenous Club Formation in a Uni-dimensional World with the

Possibility to Stay Alone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Petar Kunstek, Marko Vrdoljak Classical optimal design on annulus 156

B. Polyak Large Optimization and Asymptotic Stability . . . . . . . . . . . . . . . 159

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9

Theoretical investigation of atomic nitrogenadsorption on Si (111) surface in the framework of

molecular dynamics approach

Karine Abgaryan1, Sergey Uvarov1

1 Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

[email protected]

For many years nitridation process of silicon surface was studied [1].Today this process is applied at the initial stage of semiconductor hetero-structures growth on a substrate of silicon (111) surface by the MBE(molecular beam epitaxy) method. The nitridation process model wasproposed. That model contained chain of interdependent processes: adsor-ption of ammonia on a silicon surface, the ammonia dissociation which isfollowed by a hydrogen desorption (in the form of molecular hydrogen ormolecules of water) and by adsorption of atomic nitrogen on a surface, dif-fusion of atomic nitrogen through Si layer to the reaction front and intothe Si substrate, etc. The complete theoretical research of nitridationprocess requires research at the atomic level of all listed stages.

In this work theoretical research of one of the initial stages of nitri-dation process was conducted. Adsorption energy of atomic nitrogen invarious high-symmetric positions has been calculated on surface (111) ofsilicon using molecular dynamics approach. These calculations were usedto verify the results of molecular dynamics (MD) modeling of the Si-N sys-tem. Tersoff potential was applied with the sets of parameters received asa result of the solution of a problem of parametrical identification [2,3].Results of ab initio calculations were used as the reference values at thestage of the fitting of interatomic potential. Molecular dynamics calcula-tions of nitrogen atom adsorption at the same positions of Si(111) surfacewere performed using obtained potential. The results of these calculationswere in a good agreement with first-principles calculations. The most en-ergetically favorable position for nitrogen atom adsorption was found tobe the same via both approaches [4]. Thus, it was concluded that ob-tained potential is suitable for use in further calculations for the purposeof modeling of silicon surface nitridation process.

The authors were supported by the No. 14-11-00782 RNF.

10

References

1. Ching-Yuan Wu and Chwan-Wen King. “Growth Kinetics of silicon thermalnitridation”, Solid-state science and technology. 1982.Vol. 129, 7. 1559-1563.

2. K.K. Abgaryan, M.A. Posypkin “Optimization methods as applied to para-metric identification of interatomic potentials”, Computational Mathematicsand Mathematical Physics, 2014, 54:12, 1929-1935.

3. K.K. Abgaryan, O.V. Volodina, S.I. Uvarov “Mathematical Modeling of PointDefect Cluster Formation in Silicon Based on Molecular Dynamic Approach”,Izvestiya Vysshikh Uchebnykh Zavedenii. Materialy Elektronnoi Tekhniki =Materials of Electronics Engineering. 2015;(1):37-42.

4. K.K. Abgaryan, S.I. Uvarov, I.V. Mutigullin, Y.G. Evtushenko “Some aspectsof molecular dynamic modeling of the first stages of Si(111) surface nitridiza-tion in NH3 atmosphere”, Izvestiya Vysshikh Uchebnykh Zavedenii. Materi-aly Elektronnoi Tekhniki = Materials of Electronics Engineering. 2015;(12).

11

Approaches to the solution of the optimizationproblem of interatomic potential fitting

Karine Abgaryan1, Olga Volodina1


[email protected]

For the purpose of carrying out practical researches in the field ofmathematical simulation of crystalline structures with the given prop-erties as well as for simulation of dynamic processes, such as processesof growth of multi-layer semiconductor nanostructures, it is necessary toapply the modern potentials of interatomic interaction (Tersoff, Brenner-Tersoff, etc.). To solve the task of parametric identification for specificmaterial with the given chemical formula [1], target function of the fol-lowing form is used:

F (ξ) =

m∑

i=1

ωi

(fi(ξ)− fi

)2

→ min, ξ ∈ X, (1)

where fi – reference value of the property i, fi(ξ) – value of the same prop-erty obtained as a result of a calculation for the given set of basis atoms,ξ ∈ Rn – vector of the selected parameters, ωi – weight factor. The admis-sible set X ⊆ Rn is a parallelepiped with boundaries chosen so that it willcertainly contain possible range of parameters. It is required to define a setof the parameters ξ ∈ Rn which minimizes function F (ξ). Such a set willprovide the minimum deviation of the calculated characteristics of mate-rial from reference values, thereby it will allow to describe its propertiesmost precisely. This task becomes significantly complicated upon transi-tion from single-component (Al, Si, Fe, N, Ge, etc.) to two-componentand multicomponent materials. This complication results from the in-crease in number of optimizable parameters. Therefore the search timefor parameter sets on which the function minimum is reached increases.

It is necessary to point out that due to complication of target functionin the case of introduction of additional atomic components, it is possibleto increase uniqueness of identification of potential parameter values if thenumber of atoms in a chemical formula remains the same. In this workparameter m in (1) takes on values from one to three. The list of char-acteristics used in (1) includes cohesive energy, lattice parameter, volume

12

elastic modulus. Single-component and two-component materials wereconsidered. Methods of a zero order (a method of the Neldera-Ministryof Foreign Affairs [2], Granular Radial Search [3]) were applied. Besides,the technology of the fast automatic differentiation (FAD), described in[4] was used. It allowed us to apply the method of conjugate gradients tothe solution of the optimization task and to calculate values of the firstand second derivative composite functions with a great accuracy. The de-veloped algorithms were realized in the form of the software in languageC++. Multisequencing of computation according to input data was ap-plied. It was shown that methods of a zero order gave us a chance tofind sets of parameters leading to the minimum of function (1) with thegiven accuracy keeping the same computing expenses. The method ofconjugate gradients was tested for a case when parameters of Tersoff po-tential for single-component crystalline materials were optimized and thedietary supplement technology [3, 4] was applied. Calculations showedthat in spite of the fact that the dietary supplement technology requiresadditional arithmetic operations, however at the expense of high speedof convergence of the method of conjugate gradients it is possible to pickup sets of parameters for Tersoff potential with computing expenses com-parable with methods of a zero order. Further it is planned to applythis technology for the calculation of sets of parameters of Tersoff andBrenner-Tersoff potentials for multicomponent systems on high-perfor-mance program complexes.

The authors were supported by the No. 14-11-00782 RNF.

References

1. K.K. Abgaryan, M.A. Posypkin “Optimization methods as applied to para-metric identification of interatomic potentials”, Computational Mathematicsand Mathematical Physics, 2014, 54:12, 1929-1935.

2. A.V. Panteleev Optimization methods in examples and tasks: A.V. Panteleev,T.A. Letova - Heigh school, 2005. - 544 p.

3. D. Powell Elasticity, Lattice Dynamics and Parameterisation Techniques forthe Tersoff Potential Applied to Elemental and Type III-V Semiconductors:yew. - University of Sheffield, 2006.

4. Yu.G. Evtushenko. Optimization and fast automatic differentiation. Preprintof CCAS of Russian Academy of Sciences. 2013 of 144 p.

13

On the Cyclicity in Controllable Systems

Alexander P. Abramov1


[email protected]

We consider a dynamic process with discrete time in the form

xi(t) = ki(t)xi(t− 1), t = 1, 2, . . . ,

where x(t) = [x1(t), . . . , xn(t)] and k(t) = [k1(t), . . . , kn(t)] are positivevectors in Rn, that is, real n-dimensional space. The coefficient ki(t) iscalled an expansion rate of the variable xi(t) at the time period t. Thiscoefficient is known by the beginning of the time period. The initialvector x(0) is given.

Assume that the expansion rates are controlled by selecting the valuesof some regulating parameters. Denote by R(t) = [R1(t), . . . , RN (t)] thevector of these parameters at the time period t.

We assume that the value of Rj(t), j = 1, . . . , N , defines the coefficientrji(t) influencing the growth rate ki(t), i = 1, . . . , n, such that

ki(t) =

N∏

j=1

rji(t).

In our case, the task consists in holding the proportions xi(t)/xj(t),i, j = 1, . . . , n; i 6= j, in certain limits. This means that the followinginequalities must be satisfied

xi(t)

xj(t)6 pij , i, j = 1, . . . , n; t = 1, 2, . . . , (1)

where pij , i, j = 1, . . . , n; i 6= j, are some positive constants.Recall that a dynamic system is called controllable if there are control

actions such that the system reaches the goal. In our case, the system iscontrollable if inequalities (1) are hold for all t.

Theorem. Let the system under consideration be controllable. Sup-pose the spectrum of regulating parameter Rj, j = 1, . . . ,M , is finite.Then it is possible to use cyclically some finite sequence R(1), . . . , R(s)of regulating parameters such that inequalities (1) are hold for all t. Thissequence must be used starting from a certain time period τ0.

14

Preventive maintenance based on parametersestimation and prediction

Oleg Abramov1

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

Russia; [email protected]

For complex engineering systems under heavy-duty service the failureof which leads to heavy losses or disastrous consequences the main problemof system monitoring and diagnostics becomes not the identification andisolation of failure, but prevention of them. The solution of this task canbe based on individual preventive maintenance. Predicting and estimatingthe state of an engineering system forms an information base for individual(condition-based) maintenance.

The difficulty in solving the problem of individual status predictionis largely caused by the lack or shortage of statistic information on fieldvariation of system parameters. In this case the application of classicalmethods of mathematical statistics to the solution of status estimationand prediction problem may cause serious errors.

The paper states and solves a problem of adopting optimal estimationand prediction strategies when the stochastic properties of measurementerrors and errors of status model are unavailable. We use a technique ofindividual robust prediction which is based on the extremely propertiesof Karlin polynomials [1] and the ideas of minimax estimation. Thistechnique makes a prediction even if the number of test measurements issmall.

It does not need any stochastic properties of measurement errors andother noises (it is only necessary to know their limits), obtains not onlya simple average, but also secures bounds in which an actual value ofmeasurement parameter would lie in future. This technique has adaptiveproperties improving the prediction accuracy in an instable situation [2].

The approach under discussion meets general requirements to any pre-diction procedure. Estimates found are unique, optimal and unbiased.

In addition to measurement errors, the approach allows one to take intoaccount some other mistakes caused by the difference of real processes ofparameter variation from a mathematical model adopted. But if the basic

15

model contains an error, then a special-purpose adaptation algorithm isproposed to improve prediction accuracy.

The algorithm is based on the ideas used in the technique of movingaverage or exponential smoothing and consists in weighing measurementdata. By using prediction data we can, in optimal way, solve the problemof specifying the time of next inspection or preventive maintenance [3].

References

1. S.Karlin and W.J.Studden. “Tchebycheff Systems: with Applications in Anal-ysis and Statistics,” New York, John Wiley & Sons, (1966).

2. O.V. Abramov, A.N. Rozenbaum and A.A. Suponya. “Failure Preventionbased on Parameters Estimation and Prediction,” Preprints of 4th IFAC Sym-posium “SAFEPROCESS 2000”, vol. 1/2, 584–586, Budapest (2000).

3. O.V. Abramov and A.N. Rozenbaum. “Passive Control of the Operation ofMeasuring Instruments,” Measurement Techniques, Vol. 47, No. 3, pp. 233–239, (2004).

16

Construction of the minimal sets of differentialequations with polynomial right-hand side in

MathCloud system

A.P. Afanas’ev1, S.M. Dzyuba2, I.I. Emelyanova2,E.V. Putilina3

1 Kharkevich Institute for Information Transmission Problems RAS,

Lomonosov Moscow State University, National University of Science and

Technology “MISIS”, Moscow, Russia; [email protected] Tver State Technical University, Tver, Russia; [email protected]

3 Kharkevich Institute for Information Transmission Problems RAS, Moscow,


Let us take a look at the system of common differential equations withthe following vector notation

x = f(x), (1)

where x = (x1, . . . , xn) be a real vector function of a real variable t, andf = (f1, . . . , fn) be a real vector function, whose every element f i be amultidimensional polynomial of variables x1, . . . , xn.

Type (1) systems have long held a great deal interest for applica-tions because numerous process models of various physical, economicaland other natures are described by similar systems. The systems in ques-tion have become increasingly important as of late, since many well-knownsystems which presumably contain strange attractors belong to this par-ticular type.

Any attractor by definition is a compact invariant set. Any compactinvariant set in its turn contains a compact minimal set, described byrecurrent solution. That said, the vital problem is the construction of thewhole set of minimal sets, not separate recurrent solutions.

As of today, the question of generic minimal sets construction hasbeen raised only in publications [1,2]. However, the method of modellingminimal sets with the help of numerical techniques offers additional oppor-tunities for research into the complex behavior of dynamic systems. Thesubject of this publication is the problem of construction and explorationof minimal sets and attractors of system (1) in the distributed computer

17

environment. This construction and exploration will be based on a spe-cial method of building local solutions for the given system taking intoaccount that function f , in particular, is a multidimensional polynomial(see [2]).

The authors were supported by the Russian Science Foundation (project

16-11-10352).

References

1. A.P. Afanas’ev and S.M. Dzyuba “Method for Contraction Minimal Sets ofDynamical Systems,” Diff. Eqns, 51, No. 7, 831–837 (2015).

2. A.P. Afanas’ev and S.M. Dzyuba “Construction of the Minimal Sets of Dif-ferential Equations with Polynomial Right-Hand Side,” Diff. Eqns, 51, No.11, 1411–1419 (2015).

18

Horner’s generic scheme as applied tomultidimentional polynomials in MathCloud system

A.P. Afanas’ev1, S.M. Dzyuba2, I.I. Emelyanova2,E.V. Putilina3

1 Kharkevich Institute for Information Transmission Problems RAS,

Lomonosov Moscow State University, National University of Science and

Technology “MISIS”, Moscow, Russia; [email protected] Tver State Technical University, Tver, Russia; [email protected]

3 Kharkevich Institute for Information Transmission Problems RAS, Moscow,


Recently the problem of multidimensional polynomial evaluation hasbecome of great importance for a wide range of applications. This in par-ticular concerns the applications of optimal control, mathematical pro-gramming and qualitative theory of differential equations (e.g., see [1]).

This publication covers the development of a scheme for quick eval-uation of multidimensional polynomials. The approach is based on theHorner’s one-dimensional scheme, which allows for a relatively quick cal-culation of set values with fixed precision. The calculations are carriedout with the help of services provided by MathCloud system.

Horner’s scheme for one-dimensional polynomials is well-known andhas a lot of research dedicated to it (e.g., see [2]). Here is a short overviewof this method.

Let us consider the real polynomial

P1(x) =

n∑

i=0

aixi, (1)

where ai be an arbitrary real numbers and x be a real variable. Accordingto [2] Horner’s Scheme for polynomials (1) has the following appearance:

P1(x) = (. . . ((anx+ an−1)x + an−2)x+ . . .)x+ a0. (2)

It is easy to see that the scheme (2) allows for reducing the number ofcalculations in comparison with the direct scheme (1). Let us instantiate itis adaptation for multidimensional polynomials calculation as exemplified

19

by a two-dimensional polynomial. Specifically, let us take a look at thefollowing polynomial

P2(x) =n∑

i=0

m∑

j=0

aijxiyj, (3)

where aij be an arbitrary real numbers and x and y be a real variables.Horner’s generic scheme for polynomials (3) can be displayed as follows

P2(x) = (. . . ((bn(y)x+ an−1)x + b(y)n−2)x+ . . .)x+ b0(y), (4)

where

bi(y) = (. . . ((aim(y+ai,m−1)y+ai,m−2)y+. . .)y+ai0, i = 0, . . . , n. (5)

It is obvious that the scheme (4), (5) converts to parallels quite well.Modeling exercises has shown that using this scheme in comparison todirect scheme (3) makes it possible to reduce the number of calculationsby two or three times even in the simplest of cases still preserving thesame level of precision.

The authors were supported by the Russian Science Foundation (project

16-11-10352).

References

1. A.P. Afanas’ev. Extension of trajectories in optimal control, KomKniga,Moscow (2005).

2. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein C. Introduction to thealgorithms, MIT Press and McGraw-Hill, New York (2009).

20

Zonal Feedback Control for a Heating Problem withDelay in Boundary Conditions

Kamil Aida-zade1,2, Samir Guliyev2

1Institute of Mechanics and Mathematics of ANAS, Baku, Azerbaijan,2 Institute of Control Systems of ANAS, Baku, Azerbaijan

kamil [email protected]; [email protected]

We study the feedback control problem for objects with distributedparameters on the basis of continuous observation of the phase state of theobject at its certain points. Consider a synthesis problem of the feedbacklaw for a tubular heat exchanger with a steam jacket. The mathematicalmodel of the process can be described by the following equation [1]:

ut(x, t)+a ·ux(x, t) = −α · [u(x, t)− ϑ(t)] , (x, t) ∈ Ω = [0, l]× (0, T ]. (1)

Here a = const is the fluid velocity; l the length of the part of the heat ex-changer located inside the steam jacket; ϑ(t) the temperature of the steamjacket, which is a piecewise continuous function of time; α = const theexperimentally determined heat transfer coefficient; u(x, t) the fluid tem-perature at the point(x, t) ∈ Ω from the class of functions continuously-differentiable with respect to x and continuous with respect to t. Let theinitial and boundary conditions be given, for example, in the followingform:

u(x, 0) = u0(x, τ) ∈ G0, x ∈ [0, l], τ ∈ [−∆, 0], (2)

u(0, t) = (1− γ) · u(l, t−∆), t ∈ (0, T ]. (3)

Hereγ = γ(t) ∈ G1 = (0, δ), 0 < δ < 1, is the parameter characteriz-ing the magnitude of heat loss in the process of fluid passage throughthe heating system; ∆ the transportation delay; the continuous func-tion u0(x, τ) and the parameter γ are given inaccurately, but their valuesbelong to some given admissible sets G0 and G1, with known distribu-tion functions ρG0(u0(x, τ)) and ρG1(γ), which characterize the distribu-tion of possible values of the initial and boundary conditions. Assumethat inside the furnace, along the heat exchanger, there are installedthermocouples (sensors) measuring the fluid temperature at the pointsxj ∈ [0, l], j = 1, 2, ..., N . These measurements are used for correcting

21

the required temperature of the steam jacketϑ(t). The sensors imple-ment operational observation and input of information about the heatingprocess state at these points into the control system, as defined by thevector:

u(t) = (u(x1, t), ..., u(xN , t))∗, t ∈ (0, T ]. (4)

For the fluid heating process, it is required to synthesize the regula-tor that, based on the measurements of the temperature at the pointsxj ∈ [0, l], j = 1, 2, ..., N of the heat exchanger, would ensure the main-tenance of the output fluid temperature u(l, t) at the desired level bymanipulating the temperature ϑ(t) of the steam jacket. The consideredfeedback control problem consists in constructing the dependence of thefurnace’s temperature values from the measured state values at the ob-servational points:

ϑ(t) = w(u(t)) ∈ V, t ∈ (0, T ], (5)

minimizing the specified control quality criteria, given, for example, inthe form of the following functional:

J(w) =

∫

G0

∫

G1

T∫

0

[u(l, t;w, u0, γ)− u(t)]2dtdρG1(γ)dρG0(u0). (6)

Here u(t) is the function characterizing the desired values of the fluidtemperature at the right end (output) of the heat exchanger during thefluid heating process. For numerical solution of the feedback optimalcontrol problem (1)–(6), we propose to use the approach described in [2,3].

References

1. W.H.Ray Advanced Process Control, McGraw-Hill Inc., ChemicalEngineering,New-York (1980).

2. K.R.Aida-zade and S.Z.Guliyev Numerical Solution of Nonlinear Inverse Co-efficient Problems for Ordinary Differential Equations, J. of ComputationalMathematics and Mathematical Physics, Pleiades Publishing, vol.51, No.5,803–815 (2011).

3. K.R.Aida-zade and S.Z.Guliyev Hydraulic Resistance Coefficient Identifica-tion in Pipelines, J. ofAutomation and Remote Control, vol.77, No.7, 1225–1239 (2016).

22

On methods of minimizing a sensitivity functionunder constraints

Anatoly Antipin1


[email protected]

A system of two optimization problems is considered, the first of whichis a parametric convex programming problem with a parameter y ∈ Rm

+ ,and the second one is the problem to minimize the sensitivity of functionϕ(y), y ∈ Y ⊆ Rm

+ , generated by the first task

ϕ(y) = f(x∗) = Minf(x) | g(x) ≤ y, x ∈ X ⊆ Rn, y ∈ Rm+ , (1)

y∗ ∈ Argminϕ(y) | y ∈ Y ⊆ Rm+ . (2)

Where f(x) is scalar, and g(x) is vector function, the scalar functionand each component of the vector function are also convex, y ≥ 0, Rm

+

is positive orthant, X ⊂ Rn, Y ⊂ Rm+ is convex, closed sets. Here y is

vector right side of the functional constraints. The first problem of thesystem (1),(2) generates the sensitivity function. It is as follows: when theparameter y ∈ Rm

+ runs positive orthant, then domestic tasks in variablex ∈ X generates optimum value f(x∗), which is issued on function ϕ(y)calculated at y ∈ Rm

+ . The system (1),(2) is required find the minimumof the sensitivity functions on the set y ∈ Y , while the objective functionis defined implicitly.

The report proposes saddle-point methods for solving of system (1),(2).The convergence of computing methods is proved

This work was supported by the RFBR (project 15-01-06045), and the Pro-

gram for Support of Leading Scientific Schools (project NSc-8860.2016.1).

References

1. A.S. Antipin, E.V. Khoroshilova. “Saddle point approach to solving problemof optimal control with fixed ends,” Journal of Global Optimization, 65, No. 1,3–17 (2016).

23

Damping of oscillations of a rectangular membraneby using multiple point dampers

A.Zh. Atamuratov1, I.E. Mikhailov2, L.A. Muravey3

1 Pepsico Holding LTD, Moscow, Russia; [email protected] Dorodnicyn Computing Centre FRC CSC RAS, Moscow, Russia;

mikh [email protected] Moscow Aviation Institute (State Research University), Moscow, Russia;

l [email protected]

The oscillations of rectangular membrane are described by equation

utt = a2(uxx + uyy) + g(t, x, y), t ≥ 0, 0 ≤ x ≤ l1,0 ≤ y ≤ l2, a = const.

(1)

The initial conditions: deviation and velocity – are known:

u(0, x, y) = h0(x, y), ut(0, x, y) = h1(x, y). (2)

On the boundary of a rectangular membrane imposes fixing conditions

u(t, 0, y) = u(t, l1, y) = u(t, x, 0) = u(t, x, l2) = 0.

The problem of damping is: to find the control function g(t, x, y) ∈ L2

(0 < t < T , 0 < x < l1, 0 < y < l2), which allows to get the state of amembrane from initial state (2) to final state

u(0, x, y) = 0, ut(0, x, y) = 0. (3)

In this report we consider damping of rectangular membrane oscilla-tions by using multiple point dampers. The function g(t, x, y) is consid-ered as

g(t, x, y) =

n∑

i=1

wi(t)δ(x − xi)δ(y − yi), (4)

where wi(t) — control functions and δ — the Dirac delta-functions. Tofind the required control function numerically let us use decompositionin the finite difference scheme of (1) and coordinate descent method forminimizing of the membrane’s energy integral

E(T ) =

∫ l1

0

∫ l2

0

(u2t + a2u2x + a2u2y) dx dy.

24

Ex amp l e 1. Let us consider (1) with l1 = l2 = 1, a = 1. The initialconditions will be u(0, x, y) = sin(πx) sin(πy), ut(0, x, y) = 0, with stepshx = 0.05, hy = 0.05, ht = 0.0353. The minimum of energy integral willbe searched with precision ε = 0.001. As a function (4) we consider onlyone control function w(t) with restraint max |w(t)| = 160 placed in thepoint (x1, y1) = (0.5, 0.5). The problem was solved by T = 5.657. Figures1 and 2 show process of damping for u(t, x, 0.5) and the control functionsw(t) correspondingly.

Figure 1. The process of damping. Figure 2. The control function.

Ex amp l e 2. Let us take all conditions used in example 1. As a func-tion (5) we consider 5 equal control functions wi(t) = w(t), i = 1, ldots, 5:the one will be placed in the point as above (x1, y1) = (0.5, 0.5), whileother four will be placed in the points: (x2, y2) = (0.25, 0.25), (x3, y3) =(0.25, 0.75), (x4, y4) = (0.75, 0.25), (x5, y5) = (0.75, 0.75). The problemwas solved by T = 2.121. Figures 3 and 4 show process of damping ofrectangular membrane oscillations of u(t, x, 0.5) and the control functionsw(t) correspondingly.

Figure 3. The process of damping. Figure 4. The control functions.

By using multiple dampers approach we managed to decrease more thantwo times required minimal time for damping of rectangular membraneoscillations.

25

A benchmark of heuristic algorithms for the doubleintegrator traveling salesman problem

Artem Baklanov1, Pavel Chentsov1, Alexander Gornov2,Tatiana Zarodnyuk2

1 Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia;

[email protected], [email protected] Institute for System Dynamics and Control Theory of SB RAS, Irkutsk,

Russia; [email protected], [email protected]

As the first approximation, dynamics of some objects can be consid-ered as a linear system. Namely, it is possible to obtain a good approxima-tion of spacecraft dynamics (in deep space) by using the double integratormodel. In this regard, dynamic versions of the traveling salesman prob-lem (TSP) for controlled objects described by linear differential equationsis important. A special case of such problem is the TSP for the doubleintegrator [1].

We consider a double integrator visiting a set of given stationary pointsat a minimum travel time. Control constraints are defined in terms of aconvex compact set. We obtain an upper bound for the minimum traveltime, by developing the method of transformation of the original probleminto a generalised traveling salesman problem. This transformation isbased on a discretisation of sets of admissible visiting velocities. To solvetime-optimal two-point problems, we use the duality of optimal controlproblems and convex programming [2].

In [1] STOP-GO-STOP heuristic algorithm was proposed. It wasshown that in the worst case scenario STOP-GO-STOP provides solu-tion with total time T

√2n, where n is a number of nodes to visit and T is

the total time provided by the discretisation algorithm. Next we providea comparison of both heuristics.

Experiment setup. We generate 100 random instances of routing prob-lems. In these instances start and finish points coincide with (0, 0, 0, 0) ∈R

4. Geometric coordinates of all visiting points are drawn uniformly fromrectangle with coordinates (10,10) and (110,85). In each instance thereare exactly 14 ‘cities’ to visit. Each ‘city’ has 13 vectors of admissiblevisiting speed: (0, 0) and (4 sinα, cosα) where α = π

6 ,2π6 , ...,

12π6 . We fix

some control constraint p in 0.01 · 2n : n = 0, 1, 2, ..., 12 ⊂ [0.01, 40.96].

26

Then we apply heuristics to calculate routing time for all instances. Thesevalues are used to obtain the corresponding binary logarithm of averagecalculated time for the heuristic under constraint p. The resulting dataare depicted in Fig. 1. Note that the average calculated time is quitesimilar for p ≤ 0.16.

Fig. 1 Dependence of binary logarithm of average calculated time on control

constraint p in 0.01 · 2n : n = 0, 1, 2, ..., 12 ⊂ [0.01, 40.96] for the proposed

heuristic and STOP-GO-STOP.

The authors were supported by the Russian Foundation for Basic Research

(project no. 14-08-00419 A).

References

1. K. Savla, F. Bullo, E. Frazzoli. “Traveling Salesperson Problems for a doubleintegrator,” IEEE Trans. Automat. Contr., 54, No. 4, 788–793 (2009).

2. N.N. Krasovskii. Theory of motion control, Nauka, Moscow (1968) (In Rus-sian).

27

Experiments with Hybrid Methods ofEdgeworth-Pareto Hull Approximation in Nonlinear

Multiobjective Problems with Many Objectives

Vadim E. Berezkin1, Elizaveta A. Lotova2


[email protected] Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

[email protected]

The talk is devoted to the experience of approximating the Edgeworth-Pareto Hull, i.e. the maximal set, which Pareto frontier coincides withthe Pareto frontier of the feasible objective set, in nonlinear multiobjectiveproblems with many objectives (number of objectives is larger than 3).Approximating the Edgeworth-Pareto Hull is the most complicated stepof the Interactive Decision Maps technique, which is amed at interactivevisualization of the Pareto frontier and is an efficient tool for decision sup-port in decision problems with many objectives [1]. Graphic informationon the Pareto frontier helps the decision makers or the negotiators to spec-ify the preferred non-dominated objective point (feasible goal) consciously.Then, the associated decision is provided by the computer automatically.

In the 2000s, effective hybrid methods for nonlinear problems withnon-convex Edgeworth-Pareto Hull were developed. The hybrid methods,which integrate random search techniques and classic gradient-based op-timization methods with evolutionary multioblective techniques, turnedout to be an efficient tool for approximating the Edgeworth-Pareto Hullin non-linear problems with hundreds of decision variables and up to 9objectives. The talk is based on the paper [2].

The research was partially supported by the Program of RAS I.5 P

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. V.E. Berezkin, A.V. Lotov, and E.A. Lotova. “Study of Hybrid Methods forApproximating the Edgeworth-Pareto Hull in Nonlinear Multicriteria Opti-mization Problems,” Computational Mathematics and Mathematical Physics,54, No. 6 (2014).

28

Parallel implementation of genetic algorithm tosearch for ballistic installations optimal parameters

Nikita Bykov1,2, Nataly Vlasova2


[email protected] Bauman Moscow state technical university, Moscow, Russia;

[email protected]

The use of genetic algorithms in search for optimal parameters of bal-listic installations is becoming more widely used [1]. Optimization of theparameters of such systems places high demands on computing power dueto the fact that the solution of the direct problem is accompanied by theintegration of a two-phase non-stationary equations of gas dynamics de-scribing intrachamber processes [2]. The solution of the inverse problemof finding the optimal parameters of ballistic systems, for example, by thecriterion of maximum output speed, requires multiple solutions of the di-rect problem, since to determine the numerical value of the fitness functionis required every time to solve a system of partial differential equations,it is necessary to evaluate the quality of each chromosome. Experience inthe implementation of genetic algorithm on personal computers shows thelimitations of this approach because of the large computation time, whichdoes not allow the calculation of large populations. Fast convergence forsmall populations can not guarantee finding the global extremum. At thesame time the structure of genetic algorithm allows to make it fairly sim-ple parallelization. In this work we consider the parallel implementationof the genetic algorithm.


(project no. 16-38-00948 mol-a).

References

1. K. Li, X. Zhang ”Using NSGA-II and TOPSIS Methods for Interior BallisticOptimization Based on One-Dimensional Two-Phase Flow Model”, Propel-lants Explos. Pyrotech. 37, 468–475 (2012).

2. N.V. Bykov, E.A. Nesterenko ”Mathematical Modeling and Visualization ofIntrachamber Processes in a Ballistic Setup with Hydrodynamic Effect”, Sci-entific Visualization, 7, No. 1, 65–77 (2015).

29

Application of Random forest method to estimate theincurred but not reported claims reserve of an

insurance company

Daria Chernetsova1

1 Insurance Company Ingosstrakh Life, Moscow, Russia; [email protected]

The purpose of this report is to explore the applicability of Ran-dom forest method to assess the incurred but not reported claims reserve(IBNR) of a non-life insurance company. The research is based on thestatistical method of Random forest, presented in [1, 2], and relies on theresults in [3, 4, 5]. The actual data on the direct hull insurance of the realcompany for the period 2009-2014 were used. The following dependencewas estimated by Random forest:

paid edited ∼ crisis year of ins ev+ins sum+term end+start quar+region+claim delay, (1)

where

paid edited — sum of paid losses announced in the year next to thecontract start year, RUR,

start year — year of the policy entrance into force,

crisis year of ins ev — flag of ”crisis year of the insurance event”,

ins sum — sum insured, RUR.

term end — the term of the policy, in days,

start quar — quarter of the contract beginning date,

region — sales region of the policy,

claim delay — delay in the receipt of the loss application, calculatedfrom the contract start date, in days.

The value of the IBNR valuated on 31.12.2014 by Random forest wascompared with the results of standard calculation methods (chain ladderand Bornhuetter Ferguson on paid triangles). In general, we can say

30

that the Random forest method can be applied to assess the IBNR as analternative algorithm.

References

1. Breiman L. Random forests // Machine learning, Vol. 45, Issue 1, pp.5-32,2001.

2. Liaw A., Wiener M. Classification and Regression by Random Forest // RNews, 2(3), pp.18-22, 2002.

3. Breiman L. Out-of-bag estimation // Berkeley: Technical Report, StatisticsDepartment University of California, 1996.

4. Siroky D. Navigating Random Forests and related advances in algorithmicmodeling // Statistics Surveys, Vol. 3, pp.147-163, 2009.

5. Bylander T. Estimating generalization error on two-class datasets using out-of-bag estimates // Machine Learning, Vol. 48, pp.287-297, 2002.

31

Accelerated Primal-Dual Gradient Method forLinearly Constrained Minimization Problems

Alexey Chernov1, Pavel Dvurechensky2, Alexander Gasnikov3

1 Moscow Institute of Physics and Technology, Moscow, Russia;

[email protected] Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany,

Institute for Information Transmission Problems, Moscow, Russia;

[email protected] Moscow Institute of Physics and Technology, Moscow, Russia,

Institute for Information Transmission Problems, Moscow, Russia;

[email protected]

In this work, we consider a constrained convex optimization problemof the following form

(P1) minx∈Q⊆E

f(x) : A1x = b1, A2x ≤ b2 ,

where E is a finite-dimensional real vector space, Q is a simple closedconvex set, A1, A2 are given linear operators from E to some finite-dimensional real vector spaces H1 and H2 respectively, b1 ∈ H1, b2 ∈ H2

are given, f(x) is a ν-strongly convex function on Q with respect tosome chosen norm ‖ · ‖E on E. The last means that for any x, y ∈ Qf(y) ≥ f(x) + 〈∇f(x), y− x〉+ ν

2‖x− y‖2E , where ∇f(x) is any subgradi-ent of f(x) at x and hence is an element of the dual space E∗. Also wedenote the value of a linear function g ∈ E∗ at x ∈ E by 〈g, x〉.

Problem (P1) captures a broad set of optimization problems arising inapplications. The first example is classical entropy-linear programming(ELP) problem [1], which arises in many applications such as economet-rics, modeling in science and engineering, especially in the modeling oftraffic flows [2] and the IP traffic matrix estimation. Other examples areridge regression and elastic net approaches, which are used in machinelearning. Finally, the problem class (P1) covers problems of regularizedoptimal transport (ROT) [3] and regularized optimal partial transport(ROPT) [4], which recently have become popular in application to theimage analysis.

We extend the Fast Gradient Method [5,6] applied to the dual problemin order to make it primal-dual so that it allows not only to solve the

32

dual problem, but also to construct nearly optimal and nearly feasiblesolution to the primal problem (P1). We also equip our method witha stopping criterion which allows an online control of the quality of theapproximate primal-dual solution. Unlike [3,4,7] we provide the estimatesfor the rate of convergence in terms of the error in the solution of theprimal problem |f(xk) − Opt[P1]| and the linear constraints infeasibility‖A1xk − b1‖H1 , ‖(A2xk − b2)+‖H2 . In the contrast to the estimates in [5],our estimates do not rely on the assumption that the feasible set of thedual problem is bounded. At the same time our approach is applicablefor the wider class of problems defined by (P1) than approaches in [3,7].In the computational experiments we show that our approach allows tosolve ROT problems more efficiently than the algorithms of [3,4,7] whenthe regularization parameter is small.


(project no. 15-31-20571 mol a ved).

References

1. Fang, S.-C., Rajasekera J., Tsao H.-S. Entropy optimization and mathemat-ical programming. Kluwer’s International Series (1997).

2. Gasnikov, A. et.al. Introduction to mathematical modelling of traffic flows.Moscow, MCCME, (2013) (in russian)

3. Cuturi, M. Sinkhorn Distances:Lightspeed Computation of Optimal Trans-port // Advances in Neural Information Processing Systems, p. 2292–2300(2013)

4. Benamou, J.-D. , Carlier, G., Cuturi, M., Nenna, L., Peyre, G. IterativeBregman Projections for Regularized Transportation Problems //SIAM J.Sci. Comput., 37(2), p. A1111–A1138 (2015)

5. Nesterov, Yu. Smooth minimization of non-smooth functions. // Mathemat-ical Programming, Vol. 103, no. 1, p. 127–152 (2005)

6. Devolder, O., Glineur, F., Nesterov, Yu. First-order Methods of Smooth Con-vex Optimization with Inexact Oracle. Mathematical Programming 146(1–2),p. 37–75 (2014)

7. Gasnikov, A., Gasnikova, E., Nesterov, Yu., Chernov, A. About effectivenumerical methods to solve entropy linear programming problem // Compu-tational Mathematics and Mathematical Physics, , Vol. 56, no. 4, p. 514–524(2016)http://arxiv.org/abs/1410.7719

33

Time dilatation principle in evolutionary games ofapproach

Greta Ts. Chikrii

V.M.Glushkov Institute of Cybernetics, Kiev, Ukraine; [email protected]

Presentation is devoted to the time dilatation principle for solving thegames of approach, for which Pontryagin’s condition, lying at the heart ofall direct methods, fails. Deep insight to this condition by M.S. Nikolskijresulted in the modified condition providing construction of the pursuer’scontrol on the basis of the evader’s one in the past. Establishment ofthe close relation of the modified condition with the passage from theoriginal game with perfect information to an auxiliary game with delayedinformation led to the development of an efficient approach to solvingcomplicated games of approach.

We consider the game, whose dynamics is described by an evolution-ary system of general form, encompassing a wide range of the functional-differential systems. The gist of the approach consists in artificial wors-ening the availability of information on the current evader’s control tothe pursuer. We pass from the original game with complete informationto the game with the same dynamics and the terminal set with specialkind information delay, decreasing as the game trajectory approaches theterminal set and vanishing as the game trajectory hits the target. Thenthe obtained game with delayed information is analyzed on the basis of itsequivalence to the perfect-information game with the changed dynamicsand terminal set. The central idea of investigation is the introductionof the time dilatation function, through which the time delay function isexpressed. The time dilatation principle proves its efficiency for solvingthe problems of soft meeting for a number of second-order linear differ-ential games for which formula of the time dilatation function is deducedin explicit form. We study in detail the geometric-descriptive situationof finding ’tracks’ of the evader which provides realization of the time di-latation principle by the way of following the evader’s trajectory by thepursuer with delay in time. Sufficient conditions on the game parametersare derived insuring feasibility of the game termination both under thegeometric and the integral constraints on the players’ controls.

34

On the choice of the best forms of reinsurance

Dmitry Denisov1

1 Lomonosov Moscow State University, Russia; [email protected]

The problem of choosing the form of reinsurance is a known risk of di-viding the task and put in the form of the optimization problem. Standardnotation and concepts are as follows:

S - the total loss on the portfolio of contracts,b - The total premium of the portfolio without commission,R - The transmitted part of the loss S to the reinsurer,b(R) - Award reinsurer, including its administrative costs,Q = S −R - the total loss of the insurer,b(Q) = b− b(R) - insurer’s premium.In the framework of this problem is first dispersion model that values

R and b(R) should be selected such that the dispersion does not exceedthe self-retention of the desired level, and the expected results b(Q)−E(Q)from the net retention was as high as possible.

For the case of one insurer and one reinsurer it is formulated as aproblem

minimize b(Q)− E(Q),subject to D[Q] ≤ d,

0 ≤ Q ≤ S.The dependence structure Q of S in inequalities other than those

subordinated to additional conditions, for example, b(R) = f(E[R]) orb(R) = f(σR).

Additional conditions determine the structure of the solution Q(S).It should be noted that in addressing the problem of determining thefunction additiona solve the problem, or the second dispersion model:

minimize D[Q] ≤ d,subject to b(Q)− E(Q),

0 ≤ Q ≤ Swith the same additional conditions.

35

In this paper, the task set is solved in the original formulation, thesolutions obtained are compared with the second model.

References

1. N.L. Bowers, H.U. Gerber, J.C. Hickman, D.A. Jones, C.J. Nesbit. ActuarialMathematics. - The Society of Actuaries, 1986.

2. R.E. Beard, T. Pentikainen, E. Pesonen. Risk Theory. The stochastic Basisof Insurance (2nd ed.), London:methuen, 1977.

36

The optimal marketing strategy of the firm of aspecial type

Dmitry Denisov1, Vladislav Latiy2

1 Lomonosov Moscow State University, Moscow, Russia; [email protected] Lomonosov Moscow State University, Moscow, Russia; latiy [email protected]

We present the mathematical model of a firm selling certain product.The feature of this firm is the structure: the firm is divided into severaldistribution units (for example, department stores) each of which aimsto achieve the best sales performance in comparison with other units.Each point has its own marketing budget, approved by the head office,which can not exceed the total marketing budget. The overall aim of thecompany is fair development of all units. Hence, there is the followingproblem of the budget allocation for all units i in the set A:

pDi(c)− ci → max

ci∑i∈Aci ≤ C0

where p is the price of product, Di(c) and ci are the demand for productand commercial expenses for unit i respectively, C0 is the budget. Thus,there is a kind of competition between units for share of the budget.

The main results of this paper are 1) the proof that there is the uniquespecial solution of described problem and 2) the proof that the problemof fair marketing budget allocation is equivalent to the problem of maxi-mizing the total profit:

∑i∈A(pDi(c)− ci) → max

c=(ci)i∈A∑i∈Aci ≤ C0

.

References

1. A. von Heusinger, C. Kanzow. Optimization reformulations of the generalizedNash equilibrium problem using NikaidoIsoda-type functions // TechnicalReport, Institute of Mathematics, University of Wurzburg, Wurzburg, 2006.

2. F.M. Bass, A. Krishnamoorthy, A. Prasad, S.P. Sethi Generic and brand ad-vertising strategies in a dynamic duopoly // Marketing Science 24 (4) (2005)556-568.

37

The complexities of predicting the individualprocesses of a technical object parameters changing

at the analysis of its condition

Galina Digo1, Natalya Digo2

1 Institute for Automation and Control processes, Far Eastern Division of

Russian Academy of Sciences, Vladivostok, Russia; [email protected] Institute for Automation and Control processes, Far Eastern Division of

Russian Academy of Sciences, Vladivostok, Russia; [email protected]

The increasing number of emergencies of technogenic character at thepresent time has led to necessity of the decision of such problems as main-tenance of reliability and efficiency of difficult technical objects of respon-sible appointment. Basically such objects are created by a small number,are operated under different conditions and realize extreme technologies.It is obvious that their refusals are connected with the big material lossesor catastrophic consequences, and it is necessary to solve a problem ofreliability of concrete object, instead of reliability of type of objects. Be-sides, at the analysis of its condition is important not the number offailures, and the ability to prevent them. But the problems of optimizingthe reliability, safety and effectiveness can reasonably be considered onlyat appropriate formalization of such categories as the purpose, usefulness,losses, uncertainty, and decision-making for each concrete object. It isnecessary to consider gravity of consequences of the erroneous decisionsaccepted at all stages of creation and system operation.

The prevention of failures of technical objects for critical applicationslargely depends on the availability of monitoring and prediction of theirtechnical condition and residual resource [1]. Given their uniqueness, thestrategy of technical condition control of object on the basis of individ-ual predicting processes of object parameters change is considered as apriority.

Sense of individual prediction is that the conclusion about potentialreliability and possibility of use within a specified period of a certain spe-cific device of their total population based on observations or the valueof some chosen informative parameter [2]. The main difficulties in theindividual prediction lie in the fact that we have to work with small vol-umes of the initial data (a small set of testing results) and in the presence

38

of noise (error control), whose the statistical properties reliably are notknown. In such conditions classical methods of mathematical statisticsand the theory of random processes lose their attractive properties, andtheir use for predicting can lead to essential errors and low of the predic-tion reliability. Dominating in reliability theory probabilistic-statisticalapproach, which is based on statistics of failure, is also few suitable, sincefailures are associated with large material losses or catastrophic conse-quences. And in a considered situation speech should go not about fixingof refusals, and about their prevention.

As more productive, the functional-parametrical (functional-parame-tric) approach was offered in [3]. Within the frames of this approachfailure is a consequence of operational deviations of the parameters fromtheir initial (nominal, design) values, and the form of manifestation offailure is the output of parameters beyond the region of possible values(area of serviceability). But it has the complexities. They, in particular,are connected by that the majority systems are poorly formalized, thatdo not allow to use functional models for modeling of parameters changeprocesses, and also with deficiency of the information about the paramet-ric perturbations patterns. In the report it is analyzed, as it is possibleto overcome the difficulties arising at reduction of the considered problemto a problem of decision-making in the conditions of uncertainty for real-ization a rational way of providing of guaranteed result. For accelerationof labour-intensive processes of calculations it is offered to use multipletechnologies.

References

1. Abramov O.V. “Monitoring and forecasting of the technical condition of sys-tems of responsible appointment,” Information science and control Systems,No. 2, 4–15 (2011).

2. Piganov M.N. Individual forecasting of indicators of quality of elements andcomponents of microassembly, New technologies, Moscow (2002).

3. Abramov O.V. “The technology of prevention of failures of technical systemsof responsible appointment,” in: XII All-Russia meeting on control problemsVCPU-2014, V.A. Trapeznikov Institute of Control Sciences of the RussianAcademy of Sciences, Moscow, 2014, pp. 7540–7545.

39

Optimal control problem with state constraints

Vasily Dikusar1, Marek Wojtowicz2, Elena Zasukhina3


[email protected] Kazimierz Pulaski University of Technologies and Humanities in Radom,

Radom, Poland; [email protected] Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

[email protected]

A problem from financial mathematics is considered. The investortakes credit in bank and chooses suitable securities to get profit. Theproblem is to find a maximum return for given securities to meet the cashrequirements and liabilities.

Consider on a given compact interval [0, T ] a system of differentialequations:

x1 = u1, x2 = u2, x3 = ρ(t)x1 − ρ0(t)x2 (1)

and state constraints

x1 ≤M ; x1, x2, x4 ≥ 0; x4 = x3 + x2 − (1 + d(t))x1, (2)

where

x(t) =

x1(t)x2(t)x3(t)x4(t)

is a state vector, u(t) =

[u1(t)u2(t)

]is a control,

x1(t) is volume of securities, x2(t) is volume of credit, x3(t) is profit(return), x4(t) is cash position; u1(t) is volume rate of securities, u2(t) isvolume rate of credit; ρ0(t) is rate of credit; ρ(t) is coupon rate; d(t) isestimated risk.

The maximized value function x3(T ) could be expressed as

x3(T ) =

T∫

0

[ρ(t)x1 − ρ0(t)x2]dt. (3)

40

The control vector u(t) involves inequality constraints

u10 ≤ u1(t) ≤ u11, u20 ≤ u2(t) ≤ u21 (4)

For system (1) initial conditions are given by relations:

x1(0) = x2(0) = x3(0) = x4(0) = 0 (5)

The desired final state is:

x1(T ) = x2(T ) = 0. (6)

Denote

R(t) =

ρ0(t)ρ(t)d(t)

.

As input forecasted value of vector R(t) is needed; ρ0(t) > 0, ρ(t) > 0,0 < d(t) ≤ 1.

An approach to solution of described optimization problem is pro-posed. This approach utilizes the system dynamics and adjoint system ofdifferential equations. The resulting boundary value problem is charac-terized by Jacobi matrices which are small (in the sense of dimension) andill-conditioned. The proposed analytical solution for special cases servesas first approximation in homotopy techniques.


(project no. 15-07-08952).

References

1. A.Ya. Dubovitsky, A.A. Milutin. Necessary conditions for an extremum insome linear problems with mixed constraints, Stohastic processes and control,Nauka, Moscow (1978).

2. L.S. Pontryagin, V.G. Boltyansky, R.V. Gamkrelidze, E.F.Mishchenko Mathematical theory of optimal processes. Nauka, Moscow(1983).

3. N.D. Dikusar. “Piecwise polynimial approximation of the sixth order withautomatic knots detection,” Mathematical modelling, 22, No. 12, 115–136(2010).

4. N.D. Dikusar. “The basic element method,” Mathematical modelling, 26, No.3, 31-48 (2014).

41

Development of automated scientific informationsystem taking into account query optimization

Olga Druzhinina1, Natalia Petrova2

1 Institute of Informatics Problems, FRC CSC RAS, Russia; [email protected] Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

[email protected]

The problem of the organization of high-quality access to scientific andbibliographic information can be solved on the basis of use of the moderntechnologies of the databases (DB) allowing to structure data so that toprovide their effective collection, processing, storage and extraction [1, 2].

Some aspects of development of automated system of collection, stor-age and search of scientific information are provided in [3]. The specifiedsystem is developed on the basis of use of relational databases and annualscientific bibliographic Indexes of Dorodnicyn Computing Center of FRCComputer Science and Control. The DB conceptual model BibliographicIndexes is realized on the basis of Access DB. In an analysis result ofdata domain the following objects are selected: issuings, articles, authors,editors, reviewers, personnel, keywords, subject links, UDC. The modelconsists of a set of two-dimensional tables. For identification of entry inthe table unique primary keys are used. For communication of objects thedevice of foreign keys is used. To input personnel data we use the dividedform which simultaneously displays the data in Form mode and in Tablemode. These two modes are associated with the same data source andsynchronized with each other.

In process of query optimization in the DB search of the optimal ex-ecution plan of requests from all possible for the given request is carriedout. Query optimization is connected to change of structure of a DB forthe purpose of reduction of use of computing resources in case of exe-cution of request. The same result can be obtained in DB by differentmethods which can significantly differ both on expenses of resources, andon runtime.

In relational DB the optimal execution plan of request is sequence ofapplication of operators of relational algebra to the initial and interme-diate relations which for a specific current status of a DB (its structureand filling) can be executed with minimum use of computing resources.

42

Now the following strategy of search of the optimal plan are used: 1) byan assessment of all swaps of the connected tables, the input methodsin tables and connection types (i.e. complete search of options); 2) onthe basis of the genetic algorithm by an assessment of limited number ofswaps [4, 5].

The developed conceptual model Bibliographic Indexes can be used fordevelopment of systems of collection, storage and search of scientific andbibliographic information, for structuring bibliographic indexes of publi-cations of research teams. Use of the electronic indexr including biblio-graphic resources and links to full-text sources will allow to receive quicklyinformation that will promote increase of level and quality of activities ofscientific institution.

References

1. V.M. Moskovin. ”Databases of scientific information and on-line retrievaltools: use for control of knowledge”, Scientific and technical libraries, No. 6,18–29 (2012).

2. E.V.Fufayev, D.E. Fufayev. Databases, Academy, Moscow (2014).

3. O.V. Druzhinina O.V., A.F. Klimova., N.P. Petrova. ”Development of au-tomated system of collection, storage and search of scientific information”,Science Intensive Technologies, 17, No. 2, 4–14 (2016).

4. K. Bennett , M.C. Ferris , Y.E. Ioannidis. ”A genetic algorithm for databasequery optimization”, Proceedings of the Fourth International Conference onGenetic Algorithms, Morgan Kaufmann Publishers, Los Altos, pp. 400–407.

5. M. Stillger , M. Spiliopoulou. ”Genetic programming in database query opti-mization”, roceedings of the First Annual Conference, P. Stanford University,MIT Press, 1996, pp. 388–393. No. 12, 10–12 (2006).

43

Separation of Trivial Parts from Control Systems

Vladimir Elkin1,2


elk [email protected] Moscow Institute of Physics and Technology, Dolgoprudny, Russia

We consider nonlinear control systems that are linear in controls

y = f(y)u, y ∈M ⊂ Rn, u ∈ Rr. (1)

Here, y are the phase variables; u are the controls; and M is the phasespace that is a domain. We assume that f is an n-by-r matrix in whichthe columns fα, α = 1, 2, . . . , r, are smooth vector fields; and rankf(y) =const = r. A solution (or phase trajectory) of system (1) is definedas a continuous piecewise smooth function y(t) for which there exists apiecewise continuous control u(t) such that y(t) and u(t) satisfy (1).

Introduce two types of so-called trivial systems. The first is

y = 0.

For this system there is only one solution (which is constant) for giveninitial point y0. The second is

y = u.

For this system every continuous piecewise smooth function y(t) is solu-tion.

By separation of trivial part of the first type we mean reduction ofsystem (1) by substitution of variables to the system

z1 = 0, (2)

z2 = h(z1, z2)v, (3)

where z1, z2 — new phase variables, v — new controls.By separation of trivial part of the second type we mean reduction of

system (1) by substitution of variables to the system

z1 = v1, (4)

z2 = h(z2)v2, (5)

44

where z1, z2 — new phase variables, v1, v2 — new controls.The decomposition (2) (3) separates the trivial part (2). By analogy,

the decomposition (4) (5) separates the trivial part (4). This is help-ful in decomposing any control problem related to system (1) into twoproblems—a trivial problem related to trivial system (2) or (4) and, ingeneral, a nontrivial problem related to system (3) or (5).

For reduction to (2), (3) we must construct minimal algebra Lie ofvector fields containing vector fields fα, α = 1, 2, . . . , r [1, p. 33]. If therank of distribution generated by this algebra is equal to s < n then thedimension of z2 is equal to s and the dimension of z1 is equal to n− s. Ifs = n then reduction to (2), (3) is impossible.

For reduction to (4), (5) we must pass from (1) to dual Pfaffian systemof equations

n∑

i=1

ωki (y)dy

i = 0, k = 1, . . . , q = n− r, (6)

which can be found from (1) by eliminating the variables u and then mul-tiplying by dt. If the class of Pfaffian system of equations (6) (i.e. minimalnumber of variables on which Pfaffian system of equations equivalent to(6) may depend [1, p. 72]) is equal to s < n then the dimension of z2is equal to s and the dimension of z1 is equal to n − s. If s = n thenreduction to (4), (5) is impossible.

References

1. V.I.Elkin. Foundations of Geometric Theory of Nonlinear Control Systems[in Russian], Nauka, Moscow (2014).

45

NP-hardness of Minimum Length of Vectors SumProblems

Anton Eremeev1, Alexander Kel’manov2, Artem Pyatkin2

1 Omsk Branch of Sobolev Institute of Mathematics, Omsk State University,

Omsk, Russia; [email protected] Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk,

Russia; [email protected], [email protected]

We consider the following problems of finding a subset of Euclideanpoints (vectors) of minimum sum lengths.

Problem 1 (Subset of minimum sum length, given cardinality). Givena set Y = y1, . . . , yN of points from R

q, and a positive integer M > 1.Find: a subset C ⊂ Y of cardinality M such that

∥∥∥∑

y∈Cy∥∥∥ −→ min .

Problem 2 (Subset of minimum sum length, arbitrary cardinality).Given a set Y = y1, . . . , yN of points from R

q. Find: a non-emptysubset C ⊂ Y minimizing the same objective function.

These problems can be interpreted, in particular, as finding a balancedsubset of powers from a given set. Note that similar problems with theopposite optimization criterium (maximum) are well studied [1,2].

We prove that these problems are strongly NP-hard if the dimension qof the space is a part of the input, and NP-hard in the ordinary sense oth-erwise. Moreover, the problems are NP-hard even in the case of dimension2 (on a plane).

This work was supported by Russian Science Foundation (projects 16-11-

10041 and 15-11-10009).

References

1. A. E. Baburin, E. Kh. Gimadi, N. I. Glebov, A. V. Pyatkin “The Problemof Finding a Subset of Vectors with the Maximum Total Weight,” J. Appl.Indust. Math., 2, No. 1, 32–38 (2008).

2. A. V. Pyatkin “On complexity of finding a subset of vectors of the maximumsum length,” Discr. Anal. and Oper. Res. (in Russian), 16, No. 6. P. 68–73(2009).

46

Cost-effective covering of the strip with identicaldirectional sensors

Adil Erzin1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia;

[email protected]

Sensor networks are often used to monitor the roads, pipelines, perime-ter facilities, etc. All these objects can be modeled as a strip. When op-timizing the functioning of the sensor networks it is necessary to find anoptimal cover of the strip with various figures. To cover a strip the circles[1], ellipses [3] and sectors [2,4,5] are used [6]. If the sensor is equippedwith a video camera, then its observation area can be considered as asector [2,5]. Modern sensors can adjust the observation area [6], i.e. theparameters of the sector (angle and radius).

We study the problem of constructing a cost-effective regular coverof a strip with identical sectors. Three effective coverage models, whichwere proposed first in [5], are considered and their comparative analysisis performed which allows to obtain an upper bound for the minimumnumber of the sectors per unit length of the strip.

The author was supported by the Russian Foundation for Basic Research

(project no. 16-07-00552).

References

1. S.N. Astrakov, A.I. Erzin. “Efficient band monitoring with sensors outer po-sitioning”, Optimization, 62, No. 10, 1367–1378 (2013).

2. N. Deshpande, A. Shaligram. “Energy saving in WSN with directed connec-tivity”, Wireless Sensor Networks, 5, No. 6, 121–125 (2013).

3. A.I. Erzin, S.N. Astrakov. “Min-density stripe covering and applications insensor networks”, In: Murgante, B. et al. (eds.) ICCSA 2011. LNCS,Springer, Heidelberg, 6784, 152–162 (2011).

4. A.I. Erzin, N.A. Shabelnikova. “Sensor networks and optimal regular coveringof the plane with equal sectors”, Book Series: AER-Advances in EngineeringResearch, 15, 415–417 (2015).

5. A.I. Erzin, N.A. Shabelnikova. “On the density of a strip cover with identicalsectors”, J. of Applied and Industrial Mathematics, 9, No. 4, 461–468 (2015).

6. M.A. Guvensan, A.G. Yavuz. “On coverage issues in directional sensor net-works: A survey”, Ad Hoc Networks, 9, No. 7, 1238–1255 (2011).

47

New approach to the theorems of alternatives

Yuri G. Evtushenko1, Alexander I. Golikov1


[email protected], [email protected]

The theorems of alternatives are not only of theoretical but also ofconsiderable computational importance. They provide the opportunity ofobtaining solutions to the systems of linear equalities and inequalities, tofind the steepest descent direction in convex programming problems, toconstruct separating hyperplanes, to perform improper problems correc-tion, to create new algorythms for linear programming and so on.

The theoremes of alternatives always contain two equality and/or in-equality systems which can be expressed through I and II. The theoremesof alternatives state that any of these two systems, either I or II is al-ways soluble, but they are never soluble simultaneously. The input datafor alternative systems I and II are the elements of matrix A and thecomponents of vector b. Nevertheless alternative systems can be obtainedin another way by using different matrices of various dimentiones [1].

For a given matrix A ∈ Rm×n of rank m, let us take into considerationmatrix K ∈ Rν×n, where ν = n−m is the defect of matrix A. Any matrixcan be used as K, if ν of its rows make up the null-space basis for matrixA. Therefore AK⊤ = 0mν , where 0ij stand for (i × j)-matrix with nullelements. In selecting matrix K a sertain arbitrariness is observed, i.e. itcan be obtained in different ways. If matrix A is presented in the blockform A = [B |N ], where B is non-degenerate, then matrix K can be putas follows: K = [−N⊤(B−1)⊤| Iν ]. If, using Gauss-Jordan transform,matrix A is reduced to A = [Im |N ], then matrix K can be expressed asK = [−N⊤| Iν ]. Matrix A has m rank, hence system Ax = b is alwayssoluble, but its solutions may fail to contain non-negative ones. Let usexpress the arbitrary solution to system Ax = b as x.

Theorem. Let us asume that matrix A ∈ Rm×n has rank m, matrixK ∈ Rν×n has rank ν = n−m and AK⊤ = 0mν , vector b 6= 0m, ρ is anarbitrary positive number. Then:

1) either system

Ax = b, x ≥ 0n, (I)

48

or systemKv = 0ν , v ≥ 0n, −x⊤v = ρ > 0; (IIv)

is always consistent;2) either system

K⊤y ≤ x, (Iy)

or systemA⊤u ≤ 0n, b⊤u = ρ > 0. (II)

is always consistent.lternative systems (I) and (IIv) are similar to the corresponding al-

ternative systems in Farkas theoreme, and (Iy) and (II) — to those inGale theoreme.

Note that systems (I) and (Iy) are equivalent in the sence that theyare either simultaneously soluble or simultaneously insoluble. Likewise,systems (II) and (IIv) are equivalent in the sence of their simultaneoussolubility or insolubility.

The authors were supported by the the RFBR (project no. 14-07-00805),

by the Leading Scientific Schools Grant no. 8860.2016.1, and by the Program

of RAS I.33 P

References

1. A.I. Golikov, Yu.G. Evtushenko “On inverse linear programming problem,”Trudy Instituta Matematiki i Mekhaniki UrO RAN, no. 3. (2016).

49

Technology for a set of solutions search in one globaloptimization problem

Evgeniya Finkelstein1, Enkhbat Rentsen2, Anton Anikin1

1 Matrosov Institute for System Dynamics and Control Theory SB RAS,

Irkutsk, Russia; [email protected] National University of Mongolia, Ulaanbaatar, Mongolia

The task of finding a global solution of nonconvex finite-dimensionaloptimization problems is one of the fundamental goals of modern opti-mization. Most of the developed methods focus on organizing the searchin admissible region in a such way to expend fewer resources in unpromis-ing regions. In this case, the desired result is the best value of the function,but not all possible values of the argument, delivering extreme value.

The paper considers the problem of searching the entire set of solutionsin a nonlinear programming problem. The solution is a Nash equilibriumfor the two-dimensional four-person game, so a global extremum valueis zero. Statement of the problem contains four pairs of variables corre-sponding to the strategies of the players and four auxiliary variables thatallow one to convert game statement to a mathematical programmingproblem.

At first, a series of calculations using the MSBH method and differentstarting values allowed us to identify the fact that problem has multiple so-lutions. These solutions have three pairs of strategy-variables, acceptablevalues for the other pair of variables, auxiliary variables were also dif-ferent. Using this information as initial, the computational experimentswhen narrowing the search area was carried out. During calculations wefound out that despite the fact that the solution is not unique and be-longs to a certain curve, local methods stop at the point of admissibleregion boundary. The use of the sequential discretization method of thesearch region allowed to determine the interval, which includes the val-ues of the variables-strategies delivering function maximum, and identifya linear dependence of the auxiliary variables and define the functionalform of this dependence. For the stated problem we got the solution setdescription in general form.

The work was partly supported by the Russian Foundation for Basic Re-

search (project no. 15-07-03827).

50

Numerical study of the problem of spherical mobilerobot optimal control

Evgeniya Finkelstein1, Mikhail Svinin2, Alexander Gornov1

1 Matrosov Institute for System Dynamics and Control Theory SB RAS,

Irkutsk, Russia; [email protected] Kyushu University, Fukuoka, Japan

The problem of spherical robot control is sufficiently modern and rel-evant, the main application of such robots is a research area, i.e. it couldbe security, study of the surface or even a space. Prototypes developedby researchers in this field can be divided into two types according to theprinciple of motion. The first group includes robots, which have inside amechanism imparting a rotational moment (see eg. [1]). The second typeof mechanisms described, e.g., in [2] controls the position of the center ofmass.

The paper considers a model describing the robot, which designed asa spherical shell with the rotors (engines) inside. The robot dynamicsreduced to contact coordinate is described by the following system ofordinary differential equations:

x = G(x)J−1(x)Jr

n∑

k=1

nk(x)uk,

where state and control vectors are x , [ua, va, uo, vo, ψ]T , u , [φ1, φ2, φ3]

T ,and φi, i = 1, 3 denote the rotation angles of the engines.

The position of the contact point on the plane is given by the coordi-nates ua, va, and its coordinates on the sphere are specified by the anglesuo, vo, ψ is the auxiliary coordinate. Matrices G, J and the vectors n as aresult of sphericity of the source object contain trigonometric expressionsand determine the strongly non-linear right parts of the system.

Consider optimal control problem of transfer the robot from point x(0)to point x(T ) under the condition of minimizing the control source

J =

∫ T

0

uTudt.

The problem becomes significant more complicated, if we exclude oneof the controls (u , [φ1, φ2]

T ), i.e. leave only two rotors, which are in

51

the same plane with the axes of rotation and perpendicular to each other.Then, if the contact point falls on the equator (uo = ±(2k + 1)π2 , k =0, 1, 2, ...) the plane of the rotors placement becomes perpendicular to theplane of rolling. The system becomes locally uncontrollable, it comes toa physical singularity and requires additional settings in the calculations.

The resulting complexity is well illustrated in Fig. 1, where the solidline denotes the trajectory of the contact point on the plane for robot withthree engines and dash line is for robot with two rotors. Starting and endpoints are set as x(0) = [0, 0, 0, 0, 0] and x(10) = [0, 2, 0, 3, 0, 0, π/6]. Wesimulate horizontal position of the two rotors in the zero point, therefore,to satisfy the terminal equality ψ(5) = π/6 required loop in the line ofmotion of the plane.

Computational experiments for various end points allowed us to obtaincontrols that are physically realized, and the trajectories correspondingto the behavior of the prototype robot. Obtained controls satisfy theoptimality condition for a considered class of systems, and such criteria(minimum square control), the sum of control squares in any point of timea constant value.

The work was partly supported by the Russian Foundation for Basic Re-

search (project no. 15-07-03827).

References

2. J. Alves, J. Dias. “Journal of Systems and Control Engineering,” AdvancedMotion Control, 217, No. 6, 457–467 (2003).

3. V. Joshi, R. Banavar. “Motion analysis of a spherical mobile robot,” Robot-ica, 343–353 (2009).

52

An exact pseudopolynomial-time algorithm for aNP-hard problem of searching a family of disjoint

subsets

Alexander Galashov1, Alexander Kel’manov1,2

1 Novosibirsk State University, Novosibirsk, Russia;

[email protected] Sobolev Institute of Mathematics, Novosibirsk, Russia; [email protected]

We consider the following intractable problem [1-2].

Problem. Given a set Y = y1, . . . , yN of points from Rq and some

positive integers M1, . . . ,MJ . Find a family C1, . . . , CJ of disjoint sub-sets of Y such that

J∑

j=1

∑

y∈Cj

‖y − y(Cj)‖2 → min,

where y(Cj) = 1|Cj |

∑y∈Cj

y is the centroid (geometrical center) of the subset

Cj, under constraints |Cj| = Mj , j = 1, . . . , J , and∑J

j=1Mj ≤ N , on thecardinalities of the required subsets.

The strong NP-hardness of the problem is implied from the resultsobtained in [1], since in the cited work it was proved that the special caseof the problem when J = 1 is strongly NP-hard. In [2], a 2-approximationalgorithm which time complexity is equal to O(N2(NJ+1 + q)) was pro-posed. For the case of problem when the number J of required subsetsis fixed, this algorithm is polynomial. Currently, there are no other algo-rithmic results for the problem and the known results [3-7] were obtainedonly for its special case when J = 1. These results are described below.

It follows from [3] that one can find an exact solution in O(qN q+1)time. In [4], a 2-approximation polynomial-time algorithm of complexityO(qN2) was constructed. For the variation of the problem with an addi-tional restriction that the coordinates of the input points are integer andfor the case of fixed space dimension, in [5] an exact pseudopolynomialalgorithm was presented, which time complexity is equal to O(N(MB)q),

53

where B is the maximum absolute value of the coordinates of the in-put points. Furthermore, for the case of the fixed space dimension in [6]an FPTAS was proposed. This scheme for a given relative error ε finds(1 + ε)-approximate solution in O(N2(M/ε)q) time, that is polynomialin the size of input and 1/ε. A PTAS of complexity O(qN2/ε+1(9/ε)3/ε),where ε is a guaranteed relative error, was found in [7].

In the current work, for the variation of the problem with an addi-tional restriction that the coordinates of the input points are integer, analgorithm which finds an exact solution in O(N(N2 + qJ)(2MB+1)qJ +J2 log2N) time is constructed, where B is the maximum absolute value ofthe coordinates of the input points andM is the least common multiple forthe numbersM1, . . . ,MJ . In the case of the fixed dimension q of the spaceand of the fixed number J of required subsets, the proposed algorithm ispseudopolynomial and its time complexity is bounded by O(N3(MB)qJ ).


(projects nos. 15-01-00462, 16-07-00168) and by the grant of Presidium of RAS

(program 5, project 227).

References

1. A.V. Kel’manov, A.V. Pyatkin. “NP-Completeness of Some Problems ofChoosing a Vector Subset,” J. Applied and Industrial Mathematics, 5, No.3, 352–357 (2011).

2. A.E. Galashov, A.V. Kel’manov. “A 2-Approximate Algorithm to Solve OneProblem of the Family of Disjoint Vector Subsets,” Automation and RemoteControl, 75, No. 4, 595–606 (2014).

3. Aggarwal A., Imai H., Katoh N., Suri S. “Finding k points with minimumdiameter and related problems,” J. Algorithms, 12, 38–56 (1991).

4. Kel’manov A.V., Romanchenko S.M. “An Approximation Algorithm for Solv-ing a Problem of Search for a Vector Subset,” J. Applied and Industrial Math-ematics, 6(1), 90–96, (2012).

5. Kel’manov A.V., Romanchenko S.M. “Pseudopolynomial Algorithms for Cer-tain Computationally Hard Vector Subset and Cluster Analysis Problems,”Automation and Remote Control, 73(2), 349–354, (2012).

6. Kel’manov A.V., Romanchenko S.M. “An FPTAS for a Vector Subset SearchProblem,” J. Applied and Industrial Mathematics, 8(3), 329–336, (2014).

7. Schenmaier V.V. “An approximation scheme for a problem of search for avector subset,” J. Applied and Industrial Mathematics, 6(3), 381–386, (2012).

54

Modified Newton’s method to solve a transportationproblem

B.V. Ganin1

1 MIPT, Moscow, Russia;

Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

[email protected]

Let’s consider the transportation problem:

m∑

i=1

n∑

j=1

cijxij → min, (T )

n∑

j=1

xij = ai, 1 ≤ i ≤ m,

m∑

i=1

xij = bj , 1 ≤ j ≤ n, xij ≥ 0.

Let vector c ∈ Rmn be denoted as c = (c11c12...c1n...cm1cm2...cmn)T ,

and x ∈ Rmn be denoted as x = (x11x12...x1n...xm1xm2...xmn)T . Then a

problem (T ) can be rewritten in the form of a standard LP problem:

cT x→ min, Ax = b, x ≥ 0, (P )

where the (m+n)×mnmatrix A consists of zeros and ones so that Aij = 1when i = 1...m,m+1...n; j = ((i−1)n+1)...in, 1+kn...n+kn; k = 0...m,and Aij = 0 otherwise. Here b = (a1...am, b1...bn)

T .

It is well known from [1-3], that the problem (P ) reduces to the fol-lowing unconstrained maximization problem:

S(p, β, x) = b⊤p− 1

2‖(x+ A⊤p− βc)+‖2 → min, (1)

where β is a fixed scalar, and x is a fixed point. Then if β ≥ β∗, where β∗is a threshold value, a projection of this fixed point on the solution set ofthe primal LP problem (P ) can be found by formula:

ˆx∗ = (x+ A⊤p(β)− βc)+, (2)

55

where p(β) is a solution of (1).

It was shown in [4] that the problem (1) can be effectively solved usingthe generalized Newton’s method. But the Hessian matrix for convexpiecewise quadratic function S(p, β, x) is undefined. That’s why it wasused a generalized Hessian matrix which is (m + n) × (m + n) diagonalmatrix:

∂2pS(p, β, x) = −AD♯(z)A⊤, (3)

where D♯(z) is defined as mn × mn diagonal matrix having zi as a ithdiagonal element which is equal to 1 if (x+ A⊤p− βc)i > 0 and is equalto 0 otherwise, i = 1...mn.

When using the generalized Newton’s method the most time-consumingprocess is calculation of matrix (3). But due to the specific view of thematrix A corresponding to transportation problem the calculation of ma-trix (3) can be implemented much faster than for regular LP problemdespite the increased dimension. The structure of the matrix A consistsof four blocks, and the generalized Hessian matrix has accordingly thesame structure so it can be rewritten quite easily at each iteration. Thatmeans we actually don’t need to recalculate the multiplication of matrices(3) at every step.

The author was supported by the Program for Fundamental Researchof RAS P-15 and by the Leading scientific schools (NSh-8860.2016.1).

References

1. A.I. Golikov, Yu.G. Evtushenko. ”Solution Method for Large-Scale LinearProgramming Problems,” Doklady Mathematics, 70, No.1, 615–619, (2004).

2. 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).

3. 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

4. Mangasarian O.L. “A Newton method for linear programming,” J. of Opti-mizat. Theory and Appl., Vol. 121, pp. 1-18, (2004).

56

Adaptive method for simultaneous untangling andoptimization of 3d meshes

V.A. Garanzha1, L.N. Kudryavtseva2

Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

Moscow Institute of Physics and Technology, Moscow, Russia;[email protected], [email protected]

Untangling and construction of optimal discrete deformations for real-life problems can be highly nonlinear and stiff variational problems. Solv-ing them globally on large meshes can be prohibidly expensive. We presentnew adaptive untangling procedure based on dynamic extraction of sub-domain with tangled mesh which allows for sharp reduce of computationalcost for untangling. The idea is to extract regions where negative meshjacobian values are present and to surround them by a buffer zones whichallow for mesh untangling. During untangling procedure the number ofinverted mesh elements is reduced by a factor of 20, after that the newsubdomain is extracted and the process is repeated. We have found thateven quite stiff untangling problems require just a few global iterationsfor solution.

In the example presented in Fig. 1 mesh is untangled in four iter-ations. In this test case initial structured mesh around complex bodyis constructed using algebraic generator and contains 165 thousands of“quadrature nodes” with negative jacobian of local mapping of hexahe-dral cells. Four adaptive untangling iterations are shown. Note thatsolving the same problems globally, without extraction of subdomains isorder of magnitude more expensive. Simultaneous untangling and opti-mization means that during untangling procedure target mesh size in theboundary layer is taken into account which sharply reduces the cost ofoptimization following untangling phase.

This work is supported by the RAS Programme I.33p and by grant NSh-

8860.2016.1

57

Fig. 1. Successive mesh fragment for adaptive untangling.

58

Images and Existence of Constrained Scalar andVector Extrema

Franco Giannessi1

1 University of Pisa, Pisa, Italy;

[email protected],[email protected]

By means of the Image Space Analysis (ISA), a new necessary andsufficient condition is established for the existence of vector extrema.

In the scalar case, the sufficient part is shown to shrink to a know one.First of all, we deliver a short introduction to ISA; despite its theoreticalnature, ISA may suggest numerically viable tools; in this sense, a newresult is outlined, which could be useful for improving numerical methods,like, e.g., for escaping from a local minimum or making a discontinuitypoint to disappear.

Subsequently, we deal with the existence condition. Some examplesillustrate the results. Well known results are shown to be corollaries ofthe condition.

An application to the Bi-Level problems is then outlined. Some com-ments on perspectives end the talk.

59

On the m-Peripatetic Salesman Problem on randominputs

Edward Gimadi1,2, Alexey Istomin1, Oxana Yu. Tsidulko1,2

1 Sobolev Institute of Mathematics, Novosibirsk, Country;2 Novosibirsk State University, Novosibirsk, Russia;

[email protected], [email protected], [email protected]

Them-Peripatetic Salesman Problem (m-PSP) is a natural generaliza-tion of the classic Traveling Salesman Problem (TSP). It states as follows:given a complete undirected n-vertex graph G = (V,E) with weight func-tions wi : E → R+, i = 1, . . . ,m, the problem is to find m edge-disjointHamiltonian cyclesH1, . . . , Hm ⊂ E such that minimize or maximize their

total weight:m∑i=1

wi(Hi) =m∑i=1

∑e∈Hi

wi(e). It is known that the problem is

NP-hard. In paper [1] the asymptotically optimal approach was presentedfor the m-PSP with different weight functions on random inputs.

In this report we present an approach which under certain conditionsgives asymptotically optimal algorithms for the m-PSP with identicalweight functions on random inputs. This approach will be also correctfor the m-PSP with different weight functions.

We assume that the weights of the edges are independent and identi-cally distributed random reals, with distribution function f(x) defined on[an, bn] or [an,∞), 0 < an ≤ bn.

The approach consists of the following three steps.Step 1. We uniformly split the initial complete n-vertex graph G into

subgraphs G1, . . . Gm, so that each Gi has n vertices and about n(n−1)2m

edges.

procedure SPLIT(G):begin

for 1 ≤ i ≤ m set V (Gi) = V (G), E(Gi) = ∅;for each e ∈ E(G):

select at random with equal probabilities one of the setsE(G1), . . . , E(Gm), let it be E(Gi), add edge e to E(Gi).

end.

60

Step 2. We obtain subgraphs G1, . . . , Gm deleting all edges in Gi,1 ≤ i ≤ m, which are heavier than w∗ that selected so as to retain onlythe most light edges in subgraphs, but to have enough edges in each Gi

for Step 3.Step 3. In each subgraph Gi we build a Hamiltonian cycle, using

polynomial randomized algorithms, that w.h.p. (with probability → 1 asn → ∞) find a Hamiltonian cycle in a sparse random graph. We usealgorithms AGP [2] and AAV [3].

Theorem 1. Let the weights of the input graph be i.i.d. randomreals with uniform distribution function UNI(x) defined on [an, βn], 0 <an ≤ βn, or with shifted exponential distribution function Exp(x) = 1 −exp

(−x−an

βn

), 0 < an ≤ x , defined on [an,∞), 0 < an, with parameters

an, βn. Our approach gives asymptotically optimal solutions for the m-PSP, depending on the used algorithms [2,3]:

• m = O(n0.5−θ) , 0 < θ < 0.5, and βn/an = o(

nθ√lnn

), if we use

algorithm AGP in Step 3.

• m = O(n1−θ) , 0 < θ < 1, and βn/an = o(

nθ

lnn

), if we use algorithm

AAV in Step 3.

The authors were supported by the Russian Science Foundation (projectno. 16-11- 10041).

References

1. E.Kh. Gimadi, A.M. Istomin, I.A. Rykov, O.Yu. Tsidulko. ”Probabilisticanalysis of an approximation algorithm for the m-peripatetic salesman prob-lem on random instances unbounded from above.” Proceedings of the SteklovInstitute of Mathematics (289), No 1, 77–87 (2015).

2. E.Kh. Gimadi, V.A. Perepelitsa. ”A statistically effective algorithm forsearching a Hamiltonian contour or cycle”. Diskret. Analiz, No. 22, 15–28 (1973) (in Russian).

3. D. Angluin, L.G. Valiant. ”Fast probabilistic algorithms for Hamiltoniancircuits and matchings. Journal of Computer and System Sciences 18, No 2,155–193 (1979).

61

On Nash Point Search Algorithms for Three-PersonGames (3PG)

Evgeny Golshteyn1, Ustav Malkov2, Nikolay Sokolov3

1 CEMI RAS, Moscow, Russia; [email protected] CEMI RAS, Moscow, Russia; [email protected] CEMI RAS, Moscow, Russia; sokolov [email protected]

At OPTIMA-2014, we presented an approximate 3LP algorithm [1] for find-ing Nash points as solutions of 3PG given in the general setting. The goal ofthis talk is to determine the efficiency of 3LP algorithm for 3PG both in thegeneral [2] and in the special [3] settings.

The general setting. Assume that player l, l = 1, 2, 3, governs nl strate-gies, and a table (a

(l)ijk) determines its payoff whenever the players select strate-

gies i (1 ≤ i ≤ n1), j (1 ≤ j ≤ n2), and k (1 ≤ k ≤ n3), respectively. Thus,the finite 3PG in the mixed strategies is determined by the players’ payofffunctions fl(x) = fl(x1, x2, x3) =

∑n1i=1

∑n2j=1

∑n3k=1 a

(l)ijkx1ix2jx3k, l = 1, 2, 3,

defined over the set X = X(1) ×X(2) ×X(3) ⊂ En1+n2+n3 , where X(l) = xl =(xl1, xl2 . . . , xlnl

) ∈ Enl :∑nl

r=1 xlr = 1, xlr ≥ 0, r = 1, 2, . . . , nl, l = 1, 2, 3.

The special setting. Let Alm be an nl×nm-matrix, l, m ∈ 1, 2, 3, l 6= m.Then the players’ payoff functions fl(x) = xτ

l

∑1≤m≤3, m6=lAlmxm, l = 1, 2, 3,

x ∈ X. Here and below, τ denotes the vector’s transpose.

It turns out that solving the 3PG is tantamount to the search of the globalminimum (zero) of the Nash function N(x1, x2, x3) =

∑3l=1 δl(x), x ∈ X, where

δl(x) = maxxl∈X(l) fl(x)− fl(x), l = 1, 2, 3.

LP problems. Having fixed an arbitrary pair of strategy vectors x2 ∈ X(2),x3 ∈ X(3), we find an optimal solution x∗

1 of the following linear programmingproblem denoted as P1(x2, x3):in the general case

n1∑

i=1

(−

n2∑

j=1

n3∑

k=1

3∑

l=1

a(l)ijkx2jx3k

)x1i + α

(1)1 + α

(1)2 → min,

n1∑

i=1

( n2∑

j=1

a(3)ijkx2j

)x1i ≤ α

(1)1 , k = 1, 2, . . . , n3,

n1∑

i=1

( n3∑

k=1

a(2)ijkx3k

)x1i ≤ α

(1)2 , j = 1, 2, . . . , n2,

n1∑

i=1

x1i = 1, x1i ≥ 0, i = 1, 2, . . . , n1, α(1)1 , α

(1)2 ∈ E1;

in the special case

62

3∑

l=1

(− xτ

l

3∑

m=1, m6=l

Almxm

)+ α

(i)1 + α

(i)2 → min,

A21x1 ≤ α(1)1 en2 − A23x3, A31x1 ≤ α

(1)2 en3 − A32x2,

n1∑

i=1

x1i = 1, x1i ≥ 0, i = 1, 2, . . . , n1, α(1)1 , α

(1)2 ∈ E1,

where all nm coordinates of vectors enm are equal to 1, m ∈ 2, 3. As aconsequence, the inequality N(x∗

1, x2, x3) ≤ N(x1, x2, x3) holds for any strategyx ∈ X. Similarly, having fixed the values of the other feasible pairs of strategyvectors (x1, x3) ∈ X(1) × X(3) and (x1, x2) ∈ X(1) × X(2), we detect the opti-mal solutions x∗

2 and x∗3 for the corresponding linear programs P2(x1, x3) and

P3(x1, x2).The 3LP algorithm. Having chosen an arbitrary pair of (pure) strategy

vectors, at each iteration of the algorithm, we solve consequently three LPproblems thus obtaining a monotone non-increasing sequence Nq, q ≥ 1. Westop when the difference Nq − Nq−1 becomes small enough. If Nq equals zero(or reaches a small tolerance value), then we report a Nash point to be found.Otherwise we repeat the process starting with the next pair of strategies.

Setting Size, n1 = n2 = n3 = n Start pairs Time, minutes

General 100 163 7200 61 1873

Special 100 527 201000 352 2520

References

1. E. Golshteyn, U. Malkov, N. Sokolov. “On an approximate solution method ofa finite three-player game,” in: V Intern. Conf. on Optimization Methods andApplications (Optima-2014), Petrovac, Montenegro, Sept. 2014, pp. 84–85.

2. Ye.G. Golshtein. “An approximate method for solving finite three-persongames,” Ekonomika i matematicheskie metody, 50, No. 1, 110–116 (2014).

3. A.S. Strekalovskii, R. Enkhbat. “Polymatrix games and optimization prob-lems,” Avtomatika i telemechanika (Automation and Remote Control), No.4, 51–66 (2014).

63

Approximation algorithms for theresource-constrained project scheduling problem

Evgenii Goncharov1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia;Novosibirsk State Univercity, Novosibirsk, Russia [email protected]

We consider the resource constrained project scheduling problem with prece-dence and resource constraints (RCPSP). We are given set of activities J =1, ..., n. The partial order on the set of activities is defined by directed acyclicgraph G = (V, E). Each activity j ∈ E is characterized by its processing timepj ∈ Z+ and the resource requirement rjk(τ ) of the resource type k on the timeinterval [τ − 1, τ ), τ = 1, ..., pj . For each constrained resource type k is knownits capacity Rk

t during the time interval [t − 1, t). All resources are renewable.Activities preemptions are not allowed. The objective is to find precedence andresource feasible start times for all activities S = sj such that the makespanof the project Cmax(S) is minimized. The model of the RCPSP can be describedas follows:

Cmax(S) = maxj∈E

(sj + pj) −→ minsj

(1)

Subject to:

si + pi ≤ sj , i ∈ Pred(j), j ∈ E; (2)∑

j∈U(t)

rjk(t− sj) ≤ Rkt , k ∈M, t = 1, ..., Tk; (3)

sj ∈ Z+, j ∈ E, (4)

where Pred(j) is the set of immediate predecessor for activity j ∈ U , andU(t) = j | sj < t ≤ sj + pj is the set of the activities which are beingprocessed at the time instant [t− 1, t) at the schedule S.

Since the problem described above is NP-hard, it is reasonable to designapproximation algorithms with polynomial time complexity.

We offer few heuristic and metaheuristic algorithms for this problem. Au-thors developed deterministic [1] and stochastic [2] greedy algorithms, whichboth solve this problem with the time complexity dependant on the number ofactivities as n log n.

We also present a genetic algorithm using two variants of crossover [3]. Thiscrossovers are based on the most efficient use of limited resources. They areusing heuristics, which account for the degree of importance of resources, whichfor any instance is derived from solving the relaxed problem with accumulative

64

resources. It is known that the relaxed problem is polynomially solvable. A poly-nomial time asymptotically optimal algorithm with an absolute error tendingto zero for increasing number of activities was suggested by Gimadi [4]. We usethis approximate algorithm to solve auxiliary relaxed problem. The algorithmuses the known representation of a work list, and two decoding procedures.

Quality of these algorithms has been analyzed in computational experimentsusing the standard data sets j30, j60, and jl20 for the RCPSP from the PSPLIB-library, and numerical experiments demonstrated algorithm’s competitiveness.We have found the best solutions for a few instances from the dataset j120, andthe best average deviation from the critical path lower bound for the datasetsj60 (50000 and 500000 iterations) and j120 (500000 iterations).

The author is supported by the Russian Foundation for Basic Research(project no. 16-07-00829).

References

1. E.N.Goncharov A greedy heuristic approach for the resource-constrainedproject scheduling problem // Studia Informatica Univeralis, 2012, Vol. 9,N. 3, pp. 79-90.

2. E.N.Goncharov Stochastic greedy algorithm for the resource-constrainedproject scheduling problem // Discretnii analiz i issledovanie operatsii.Novosibirsk: Institut matematiki. 2014, Vol. 12, N 3, P. 10–23 (in Russian).

3. E.N.Goncharov A genetic algorithm for the resource constrained projectscheduling problem // Automation and Remote Control. Moscow, 2016, tobe printed.

4. E.Kh.Gimadi On some mathematical models and methods of planning oflarge-scale projects // Models and methods of optimization: Trudy AkademiiNauk SSSR, Siberian branch. Institut matematiki. Novosibirsk: Nauka, 1988.Vol. 10, P. 89–115 (in Russian)

65

On Some Shape Optimization Problems for ThinPlates

Vasily Goncharov1, Leonid Muravey1, Elchin Eyniev1

1 Moscow Aviation Institute (National Research University), Moscow, Russia;[email protected]; [email protected]

This work is devoted to the study of shape optimization problems for thinelastic plates. The results presented here are a natural continuation of [1–4].Let Ω be a non-empty bounded domain in R

2. Consider the following boundary-value problem: find y such that

∆ [D(x, u(x))∆y] = q, x = (x1, x2) ∈ Ω, (1)

y∣∣Γ1

= ∆y∣∣Γ1

= 0, y∣∣Γ2

=∂y

∂ν

∣∣∣∣Γ2

= 0, (2)

where ∂Ω = Γ1 ∪ Γ2, Γ1 ∩ Γ2 = ∅, D(x, u(x)) = e(x)uα(x), α > 0, u ∈ U ,

U =

u ∈ L∞(Ω) : 0 < h ≤ u(x) ≤ h,

∫

Ω

ρ(x)u(x)dx ≤ m

. (3)

For a transversely loaded plate, the equation describing the equilibrium of theplate has the form (1), and the boundary conditions imposed at y express thefact that the plate is supported on a part Γ1 of its edge and clamped on theremaining part Γ2. Let y[u] denote a solution of (1), (2), corresponding tou ∈ U . Consider the following minimization problem:

F [u] =

∫

Ω

f(x, y[u],∇y[u],∆y[u]) dx→ min, u ∈ U . (4)

Theorem 1. Let the integral functional

L2(Ω)×[L2(Ω)

]2 × L2(Ω) ∋ (y1, y2, y3) 7→∫

Ω

f(x, y1, y2, y3) dx

be lower semicontinuous with respect to the strong convergence of (y1, y2) in

L2(Ω) ×[L2(Ω)

]2and the weak convergence of y3 in L2(Ω). Then there exists

a solution to the problem (4) determined by (1)–(3).Some special cases of this theorem were considered in [5].The equation describing the buckling of a three-layered plate is as follows:

∆(e(x)u(x)∆y) = λ∆y. (5)

66

Consider the optimization problem

λ1[u] → max, u ∈ U , (6)

where λ1[u] denotes the lowest eigenvalue of (5), corresponding to u ∈ U .Theorem 2. There exists a solution u to the problem (6) determined by

(2), (3), (5). Moreover, there exist an eigenfunction y associated with λ1[u] anda number γ such that the following pointwise conditions hold:

|∆y(x)| < γ ⇔ u(x) = h,

|∆y(x)| = γ ⇐ h < u(x) < h,

|∆y(x)| > γ ⇔ u(x) = h.

These results as well as another obtained ones will be presented in the talkin detail.

The authors were supported by the Russian Foundation for Basic Research(project no. 16-01-00425).

References

1. I.E. Mikhailov, L.A. Muravey “On Control with Coefficients for High OrderPartial Differential Equations”, in: Proceedings of the III International Con-ference on Optimization Methods and Applications “Optimization and Appli-cations” (OPTIMA-2012), Dorodnicyn Computing Centre of RAS, Moscow,2012, pp. 170–174.

2. V.Yu. Goncharov, L.A. Muravey, V.M. Petrov. “On Design of a VibratingBeam on an Elastic Foundation for the Maximum Fundamental Frequency”,in: Proceedings of the V International Conference on Optimization Methodsand Applications “Optimization and Applications” (OPTIMA-2014), Dorod-nicyn Computing Centre of RAS, Moscow, 2014, pp. 86–87.

3. V.Yu. Goncharov. “Existence Criteria in Some Extremum Problems InvolvingEigenvalues of Elliptic Operators”, Journal of Siberian Federal University.Mathematics & Physics, 9, No. 1, 37–47 (2016).

4. E.T. Eyniev. “A parametric optimization method for the Dirichlet problem”,Journal of Instrument Engineering, No. 7, 47–51 (1994). (in Russian)

5. P. Neittaanmaki, J. Sprekels, D. Tiba. Optimization of Elliptic Systems,Springer, New York (2006).

67

Applied optimization problems: How to treat them

Alexander Gornov, Alexander Tyatyushkin, Tatiana Zarodnyuk,Anton Anikin, Evgeniya Finkelstein

Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk,Russia; [email protected]

Theory and practice of creation the algorithms and software for optimiza-tion problems has achieved considerable success in recent decades in Russia andabroad. There were developed a large number of reliable optimization software.It includes algorithms and tools, which allow to design effective technology, tomonitor the computational process, to verify the solution and visualize calcula-tion results.

However, the practical application of this wide range of software, unfortu-nately, is going too slow. In our opinion, the main reason is the complexity andpoor formalization of mathematical simulation methods as well as the complexsystem of business relations, including information exchange and verification ofthe results of ”temporary creative collectives”, formed to solve specific appliedproblems.

The report discusses problems arising when one applies optimization toolsto solve meaningful extremal problems. We propose the classification of theproblems according to the criteria of dimension, complexity and costs and for-mulate the software requirements. The methods of estimation of models qualityand checking their adequacy are considered. Suggested multimethod computingalgorithms allow to increase the probability of success. Developed by authorssoftware was applied for solving real problems from various scientific areas, suchas mechanics, flight dynamics, cosmonavigation, electricity, robotics, economics,ecology, medicine, nanophysics, seismology and others.

The work was partly supported by the Russian Foundation for Basic Re-search (project no. 15-07-03827 and no. 15-37-20265.

References

1. Gornov A.Yu. Computational technologies for solving optimal control prob-lems, Novosibirsk, Nauka (2009)

3. Tyatyushkin A. I. Multimethod optimization of controllable systems, LAPLAMBERT Academic Publishing (2013).

68

An Approach to Fractional Optimization

Tatiana Gruzdeva1, Alexander Strekalovsky1

1 Matrosov Institute for System Dynamics and Control Theory of SB RAS,Irkutsk, Russia; gruzdeva,[email protected]

The paper addresses the development of efficient methods for fractionalprogramming problems [1] as follows

(P) f(x) :=m∑

i=1

ψi(x)

φi(x)↓ min

x, x ∈ S,

where φi(x) > 0, ψi(x) > 0, i = 1, . . . ,m, ∀x ∈ S. This is a noncon-vex problem with multiple local extremum which belongs to a class of globaloptimization. Together with problem (P) we will also consider the followingparametric optimization problem

(Pα) Φα(x)= Φ(x, α) :=

m∑

i=1

[ψi(x)− αiφi(x)] ↓ minx, x ∈ S,

where α = (α1, . . . , αm)⊤ ∈ IRm is the vectorial parameter.Let us introduce then the optimal value function V(α) of Problem (Pα) as

followsV(α) := inf

xΦα(x) | x ∈ S.

In addition, suppose that the following assumptions are fulfilled:

(H1)

(a) V(α) > −∞ ∀α ∈ K,where K is a convex set from IRm;(b) ∀α ∈ K ⊂ IRm there exists a solution z = z(α) to

Problem (Pα), i.e. V(α) =m∑i=1

[ψi(z)− αiφi(z)].

Then it takes place the reduction (equivalence) theorem for the fractionalprogramming problem with d.c. functions and the solution of the equationV(α) = 0 with the vector variable α = (α1, . . . , αm)T satisfying the followingnonnegativity assumption

(H(α)) ψi(x)− αiφi(x) ≥ 0 ∀x ∈ S, i = 1, . . . ,m.

Theorem. Suppose that in Problem (P) the assumptions (H1) are fulfilled.In addition, let there exist a vector α0 = (α01, . . . , α0m)⊤ ∈ K ⊂ IRm for whichthe assumption (H(α0)) is satisfied.

69

Besides, suppose that in Problem (Pα0) the following equality holds

V(α0)= min

x

m∑

i=1

[ψi(x)− α0iφi(x)] | x ∈ S

= 0.

Then any solution z = z(α0) to Problem (Pα0) turns out to be a solution toProblem (P), so that z ∈ Sol(Pα0) ⊂ Sol(P).

This theorem opens the door to a justified use of the Dinkelbach’s approachfor solving fractional programming problems with the goal function presentedby a sum of fractions all given by d.c. functions.

Therefore, instead of solving Problem (P) we propose to combine a solvingProblem (Pα) with a search with respect to parameter (α ∈ IRm

+ ) in order tofind a vector (α0 ∈ IRm

+ ) such that V(α0) = V(Pα0) = 0.In this situation for every α ∈ IRm

+ we must be able to find a global solutionto (Pα) and we can do it using the global search theory [2].

Finally, computational simulation testings have been carried out for somespecial test functions formed by linear and/or convex quadratic functions. First,the experiments have been performed on the low dimension’s examples from [3].Afterwords, the approach has been tested on specially designed test problemsup to dimension n = m = 100. At the end, the test problems of dimensionup to n = m = 200 designed with the help of [4] have been also solved by thedeveloped algorithm.

This research is supported by the Russian Science Foundation (grant 15-11-20015).

References

1. J. B. G. Frenk, S. Schaible. “Fractional programming,” in: Handbook ofGeneralized Convexity and Generalized Monotonicity, Series Nonconvex Opti-mization and Its Applications, V. 76, Springer, 2002, pp. 335–386.

2. A.S. Strekalovsky. Elements of nonconvex optimization, Nauka, Novosibirsk(2003) (in Russian).

3. B. Ma, L. Geng, J. Yin, L. Fan. “An effective algorithm for globally solvinga class of linear fractional programming problem,” Journal of software, 8, No.1, 118–125 (2013).

4. Y.-C. Jong “An eficient global optimization algorithm for nonlinear sum-of-ratios problem,” Repository of e-prints about optimization and related topics,2012 (http://www.optimization-online.org/DB FILE/2012/08/3586.pdf)

70

Determination of optimal forcing directions forsynchronization of nonlinear oscillations

Vladimir Jacimovic1, Nikola Konatar1

1 Faculty of Mathematics and Natural Sciences, University of Montenegro,Dorda Vaingtona bb, 81000 Podgorica, [email protected]; [email protected]

The ability of oscillators to synchronize their oscillations is a fascinatingphenomena with great variety of manifestations important in Physics and Lifesciences ([?, ?]). In almost all fields of engineering this phenomena is takenadvantage of to modify nonlinear oscillations using a periodic external signal.

Forced synchronization is traditionally understood as a modification of self-sustained oscillations to a weak periodic signal with approximately the same (orresonant) frequency. In this way even a very weak periodic external signal can”entrain” nonlinear oscillator to oscillate at the same frequency as the forcingsignal. This adjustment of self-sustained oscillations to weak periodic force iscalled frequency locking.

However, if the external signal is rather strong, then synchronization ofself-sustained oscillations can occur even if their frequency is not close (norresonant) to the forcing frequency. This effect is called suppression of self-sustained oscillations.

We treat the problem of forced frequency locking as a bifurcation controlproblem. Nonlinear oscillators are described as dynamical systems near fixedpoint oscillatory bifurcation, while external forcing is treated as a weak control,or perturbation.

We present a mathematical result of finding optimal forcing directions in or-der to synchronize oscillations to external signal. This result can be especiallyrelevant for systems exhibiting complex dynamics that is sensitive to small vari-ations of the initial point and/or system parameters. We show our results onillustrative examples.

71

Some Generalizations of Gradient-type ProjectionMethod for Solving Quasi Variational Inequalities

Milojica Jacimovic1, Nevena Mijajlovic1, Muhammad Aslam Noor2

1 University of Montenegro, Podgorica, Montenegro;[email protected];[email protected]

2 COMSATS Institute of Information Technology, Islamabad, Pakistan;[email protected]

The theory and methods for solving variational inequalities are thoroughlytreated in the scientific literature. An important generalization of variationalinequalities are quasi-variational inequalities. If the convex set, which involvedin the variational inequality, also depends upon the solution then the variationalinequality is called the quasi variational inequality.

We study the following quasi variational inequality: find x∗ ∈ C(x∗) suchthat

〈F (x∗), y − x∗〉 ≥ 0 ∀y ∈ C(x∗), (1)

where C : H → 2H is set-valued mapping with nonempty convex and closed setC(x) ⊆ H for all x from Hilbert space H.

Note that the difficulty of problems with quasi variational inequalities isrelated to the fact that one must simultaneously solve a variational inequalityand calculate a fixed point of a set-valued mapping. This explains why the liter-ature on solution methods for quasi variational inequalities is not too extensive.Consequently, there are numerous open questions.

Projection methods represent an important tool for finding the approximatesolution of various types of variational inequalities. These methods have beenextended and modified in various ways. The main idea in these techniques isto establish the equivalence between the quasi variational inequalities and somefixed point problem. We use this alternative equivalent formulation to suggestand analyze two variants (one-step and two-step) of iterative projection methodfor solving quasi variational inequalities.

Algorithm 1. Let x0 ∈ H be an arbitrary initial approximation of thesolution. Suppose that, for a cerain k ≥ 0, the approximation xk ∈ C(xk) hasalready been determined. Then the set C(xk) is defined. Find xk+1 by

xk+1 = (1− ak)xk + akPC(xk)[xk − αF (xk)], k = 0, 1, . . .

where parameters 0 < ak ≤ 1 and α > 0 can be choosen on diferent ways.

72

Let us remark that fixed point formulation of problem (1) can be written as

u = PC(x)[x− αF (x)],

x = PC(u)[u− αF (u)].

This formulation enables to suggest and analyze the following two-step methodfor solving quasi-variational inequality (1).

Algorithm 2. For a given initial point x0 ∈ H , find the approximatesolution xk+1 by the iterative schemes

uk = (1− bk)xk + bkPC(xk)[xk − αF (xk)],

xk+1 = (1− ak)xk + akPC(uk)[uk − αF (uk)], k = 0, 1, 2, . . .

where 0 < ak ≤ 1, 0 ≤ bk ≤ 1 for all k ≥ 0 and α > 0 are parameters of method.This method is quite different than Koperlevich method. In case of bk = 0 forall k ≥ 0, this method is described by Algorithm 1.

Important class of methods for solving the quasi variational inequalities isalso a class of continuous method, in which the process is described by differen-tial equations. For the calculation of solutions of quasi-variational inequalities,we construct trajectories that start at an arbitrary point, and during the timeconverge to the set of solutions:

x′(t) = −a(t)x(t) + a(t)PC(x(t))[x(t)− αF (x(t))], x(0) = x0,

where x0 is given initial point in H and a(t) and α are parameters of the method.For a(t) ≡ 1, this method becomes standard continuous gradient-type method.

We also establish sufficient conditions for the convergence of the proposedmethods and estimate the rates of convergence.

73

Lower Bound on Restricted Isometry Constants forTight Frames

Igor Kaporin1

1 Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;[email protected]

Tight frames are the key computational tool in such areas as compressedsensing [1] and other data communication technoligies. The quality of a tightframe (represented by rectangular m × n matrix A possessing the tightnessproperty AAH = n

mIm) is characterized by the standard restricted isometry

property (RIP)

∃ δ = δ(m,n)(k) ∈ (0, 1) ∀v : ‖v‖0 = k ⇒ 1− δ ≤ ‖Av‖2‖v‖2 ≤ 1 + δ,

where ‖v‖0 denotes the number of nonzero components in v. It is implied that1 ≪ m≪ n, and for a possibly large k it is desirable to have a possibly smallerrestricted isometry constant δ. The latter is obviously related to the commonbound on condition numbers over all m×k submatrices AJ = [aj(1), . . . , aj(k)] ofA. Recall that the evaluation of the RIP constant is an NP-hard problem, see [1]and references therein. Our main result establishes the following interior boundsfor the extreme eigenvalues of AH

J AJ over all index subsets J = j(1), . . . , j(k)holding for any tight frame:

min|J|=k

λmin(AHJ AJ ) ≤ ρ

(m,n)min , max

|J|=kλmax(A

HJ AJ) ≥ ρ(m,n)

max ,

where ρ(m,n)min and ρ

(m,n)max are the smallest and largest roots of the kth degree

polynomial in λ of the form P(m−k,n−m−k)k

(1− 2m

nλ), respectively. Here

P(α,β)k (x) =

k∑

q=0

(k+α

q

)(k+β

k−q

)(x− 1)k−q(x+ 1)q (1)

is the standard kth degree Jacobi polynomial [2]. The proof is based on thefollowing determinant identity valid for any tight frame:

∑

|J|=k

det(AHJ AJ − λIk) =

( n

2m

)k

P(m−k,n−m−k)k

(1− 2m

nλ

). (2)

Addressing the lower spectral bound, one can notice that if

λ∗ = min|J|=k

λmin(AHJ AJ),

74

then the left hand side of (2) is nonnegative for all λ ∈ [0, λ∗], and thereforeits right hand side cannot be negative, which yields, by the simplicity of Jacobipolynomial roots,

λ∗ ≤ ρ(m,n)min .

Similarly, one can obtain the bound for the largest eigenvalues of AHJ AJ .

Moreover, considering the quantity

δ(k,m, n) =

√(1

m− 1

n

)(k − 1

4

)log

(k − 1

4

),

one can verify numerically that for all 2 ≤ k ≤ m/2,√n ≤ m ≤ n/2, n ≤ 256,

and under a natural restriction δ(k,m, n) ≤ 1/√2, it holds

min|J|=k

λmin(AHJ AJ ) ≤ ρ

(m,n)min ≈ 1− δ(k,m, n),

max|J|=k

λmax(AHJ AJ) ≥ ρ(m,n)

max ≈ 1 + δ(k,m,n).

This observation well agrees with the widely known probabilistic estimates ofthe restricted isometry in stochastic rectangular matrices, cf.[1]. It must bestressed that the presented result is completely deterministic.

The author was supported by the Russian Foundation for Basic Research(project no. 14-07-00805), by Fundamental Research Program of R.A.S. no. I.5 Π,and by the grant NSh-4640.2014.1.

References

1. Foucart S., Rauhut H. A mathematical introduction to compressive sensing(Basel: Birkhauser, 2013)

2. Bateman H., Erdelyi A. Higher Transcendental Functions. Bessel Functions,Parabolic Cylinder Functions, Orthogonal Polynomials (New York: McGraw-Hill,1953)

75

Generalized solutions of optimal control problems

Dmitry Karamzin1, Fernando Pereira2

1 Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;dmitry [email protected]

2 University of Porto, Porto, Portugal; [email protected]

Not all the problems of the classical calculus of variations have a solution.The following problem is, perhaps, one of the simplest examples to confirm thisthesis.

Minimize

∫ 1

0

x2(t)dt

subject to x(0) = 0, x(1) = 1.

(1)

Here, the minimum of integral is sought on the set of all smooth functionsx(t), whose values in points t = 0, 1 are fixed. A classic smooth solution toproblem (1) does not exist. Indeed, there does not exist even a continuoussolution: any minimizing sequence of arcs from example (1) pointwise convergesto a discontinuous function y(t) such that y(t) = 0 when t ∈ [0, 1), and y(1) = 1.

Considering more general optimal control problems, the situation becomeseven more complicated by the following reason. There are examples of differ-ential control systems with terminal constraints, for which there does not existeven a single continuous admissible (feasible) arc. Indeed, consider the controlsystem

x = u, y = 1− x, u ≥ 0, t ∈ [0, 1], (2)

x(0) = 0, x(1) = 1, y(0) = 0, y(1) = 0.

It is clear that x(t) is increasing, but x(t) ≤ 1. Therefore, y(t) is increasingas well. Thereby, in view of the terminal constraints, the compatibility of thissystem is possible only if y(t) ≡ 0. Hence, x(t) = 1 ∀ t ∈ (0, 1], and then atpoint t = 0, function x(t) is discontinuous. By considering any functional over(2) we obviously come to a problem, in which continuous solutions do not exist.

The absence of classical solutions naturally gives rise to the so-called exten-sion or relaxation of the problem of Variational Calculus or Optimal Controland leads to the notion of a generalized solution. Under the “extension” it isbeing understood the introducing into consideration the so-called generalizedsolutions: it is necessary to relax the notion of arc, that is to enlarge the classof admissible functions x(t), so that in the enlarged class of arcs solutions to(1), (2) would already exist.

76

In this work, we construct an extension for general nonlinear control problem

Minimize φ(x0, x1)

subject to x = f(x, u, t),x0 = x(0) ∈ A, x1 = x(1) ∈ B,u(t) ∈ U a.a. t ∈ [0, 1],

(3)

where φ, f – given smooth functions, A,B,U – given closed sets, and u(t) – ameasurable function (for example, of class Lp([0, 1]), p ≥ 1).

Discontinuous trajectories can be expected only when the set U is un-bounded. Indeed, on the one hand, it is clear that discontinuities occur whenthe arc derivative begins to take unbounded values (which is possible only whenthe set U is unbounded), see Example (1). On the other hand, in the case ofbounded set U , and under fairly general assumptions an extension of (3) intothe class of absolutely continuous arcs is still feasible. Such an extension wasproposed in [1] and is based on the notion of a generalized control.

In our investigation we combine the methods of [1] with the method of so-called Lebesgue discontinuous time variable change resulting the generalizedimpulsive control.

The authors were supported by the Russian Foundation for Basic Research(project no. 16-31-60005), and by FCT (Portugal) research funding grantedto the SYSTEC R&D Unit under project UID/EEA/00147/2013, and projectNORTE-01-0145-FEDER-000033 – STRIDE (COMPETE 2020).

References

1. R.V. Gamkrelidze Principles of Optimal Control theory, Plenum Press, New-York, 1978.

77

On some clustering problems: NP-hardness andefficient algorithms with performance guarantees

Alexander Kel’manov1

1 Sobolev Institute of Mathematics, Novosibirsk State University,Novosibirsk, Russia; [email protected]

We consider some quadratic Euclidean clustering problems most closely re-lated to data mining, machine learning, statistics, and computational geometry.The report purpose is review of some new results on the computational complex-ity of these problems, and on efficient algorithms with performance guaranteesfor their solutions.

Below is a list of considered problems.

Problem 1. Given a set Y = y1, . . . , yN of points from Rq and some

positive integers M1, . . . ,MJ . Find a family C1, . . . , CJ of disjoint subsets ofY such that

J∑

j=1

∑

y∈Cj

‖y − y(Cj)‖2 → min,

where y(Cj) is the centroid (geometrical center) of the subset Cj, under con-straints |Cj | =Mj , j = 1, . . . , J, and

∑Jj=1Mj ≤ N .

Problem 2. Given a set Y = y1, . . . , yN of points from Rq and a positive

integer M . Find a partition of Y into two non-empty clusters C and Y \ C suchthat

|C|∑

y∈C

‖y − y(C)‖2 + |Y \ C|∑

y∈Y\C

‖y‖2 −→ min

where y(C) is the centroid of C, subject to constraint |C| =M .

Problem 3. Given a sequence Y = (y1, . . . , yN) of points from Rq,

and some positive integer numbers Tmin, Tmax and M . Find a subset M =n1, . . . , nM of N = 1, . . . , N such that

∑

j∈M

‖yj − y(M)‖2 → min,

where y(M) is the centroid of yj | j ∈ M, under constraints

Tmin ≤ nm − nm−1 ≤ Tmax ≤ N, m = 2, . . . ,M, (1)

on the elements of (n1, . . . , nM ).

78

Problem 4. Given a sequence Y = (y1, . . . , yN ) of points from Rq and

some positive integers Tmin, Tmax, L, and M . Find nonempty disjoint subsetsM1, . . . ,ML of N = 1, . . . , N such that

L∑

l=1

∑

j∈Ml

‖yj − y(Ml)‖2 +∑

i∈N\M

‖yi‖2 → min, (2)

where M =⋃L

l=1 Ml, and y(Ml) is the centroid of subset yj | j ∈ Ml, underthe following constraints: (i) the cardinality of M is equal to M , (ii) concate-nation of elements of subsets M1, . . . ,ML is an increasing sequence, providedthat the elements of each subset are in ascending order, (iii) the inequalities (1)for the elements of M = n1, . . . , nM are satisfied.

Problem 5. Given a sequence Y = (y1, . . . , yN) of points from Rq

and some positive integers Tmin, Tmax, and L. Find nonempty disjoint sub-sets M1, . . . ,ML of N = 1, . . . , N such that the objective function (2) wouldbe minimal, under the following constraints: (i) concatenation of elements ofsubsets M1, . . . ,ML is an increasing sequence, provided that the elements ofeach subset are in ascending order, (ii) the inequalities (1) for the elementsof M = n1, . . . , nM are satisfied; (the cardinality of M assumed to be un-known).

Problem 6. Given a set Y = y1, . . . , yN of points from Rq, and positive

integer J > 1. Find: a partition of Y into subsets C1, . . . , CJ such that

J∑

j=1

1

|Cj |∥∥∥∑

y∈Cj

y∥∥∥2

−→ min .

The study of problems 1–3 and 5 was supported by the the Russian Founda-tion for Basic Research (projects no. 15-01-00462, no. 16-07-00168). The studyof problems 4 and 6 was supported by the Russian Science Foundation (projectno. 16-11-10041).

79

An approximation algorithm for one NP-hardproblem of partitioning a sequence into clusters with

restrictions on their cardinalities

Alexander Kel’manov1,2, Sergey Khamidullin1, VladimirKhandeev1,2, Ludmila Mikhailova1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia;2 Novosibirsk State University, Novosibirsk, Russia;

kelm,kham,khandeev,[email protected]

We consider the following strongly NP-hard [1]Problem. Given a sequence Y = (y1, . . . , yN) of points from R

q andsome positive integers Tmin, Tmax, L, and M . Find nonempty disjoint subsetsM1, . . . ,ML of N = 1, . . . , N such that

F (M1, . . . ,ML) =L∑

l=1

∑

j∈Ml

‖yj − y(Ml)‖2 +∑

i∈N\M

‖yi‖2 → min, (1)

where M = ∪Ll=1Ml, and y(Ml) = 1

|Ml|

∑j∈Ml

yj yj | j ∈ Ml, under the

following constraints: (1) the cardinality of M is equal to M , (2) concatenationof elements of subsets M1, . . . ,ML is an increasing sequence, provided that theelements of each subset are in ascending order, (3) the following inequalities forthe elements of M = n1, . . . , nM are satisfied:

Tmin ≤ nm − nm−1 ≤ Tmax ≤ N, m = 2, . . . ,M.

At present, for Problem 1, except for its particular case when L = 1 in (1),there are no efficient algorithms with guaranteed accuracy. For the mentionedcase of Problem 1 the following results were obtained.

In [1], the variant of Problem 1 in which Tmin and Tmax are the parameterswas analyzed. In the cited work it was shown that in the case when L = 1 thisparameterized variant is a strongly NP-hard problem for any Tmin < Tmax. Inthe trivial case when Tmin = Tmax the problem is solvable in polynomial time.

In [2], for the same case of Problem 1, when L = 1, a 2-approximationpolynomial-time algorithm having O(N2(MN+q)) running time was presented.

In addition, in [3,4], two subcases for the same case of problem when L = 1were studied. In both subcases the dimension q of the space was fixed. In [3],

80

for the subcase with integer inputs an exact pseudopolynomial algorithm wasconstructed. The time complexity of this algorithm is O(MN2(MD)q), whereD is the maximum absolute coordinate value in the input set of points. For thesubcase with real inputs in [4] a fully polynomial-time approximation schemewas proposed which, given a relative error ε, finds a (1+ε)-approximate solutionof Problem 1 in O(MN3(1/ε)q/2) time.

The main result of this paper is an algorithm that allows to find a 2-approximate solution of Problem 1 in O(LNL+1(MN + q)) time, which is poly-nomial if the number L of clusters is fixed.

The authors were supported by the Russian Science Foundation, project no.16-11-10041.

References

1. A.V. Kel’manov, A.V. Pyatkin. “On Complexity of Some Problems of ClusterAnalysis of Vector Sequences,” J. Appl. Indust. Math., 7, No. 3, 363–369(2013).

2. A.V. Kel’manov, S.A. Khamidullin. “An Approximating Polynomial Algo-rithm for a Sequence Partitioning Problem,” J. Appl. Indust. Math., 8, No.2, 236–244 (2014).

3. A.V. Kel’manov, S.A. Khamidullin, V.I. Khandeev. “An Exact Pseudopoly-nomial Algorithm for a Sequence Bi-Clustering Problem,” Book of abstractsof the XVth Russian Conference ”Mathematical programming and applica-tions”, Ekaterinburg, 139–140. (2015). (in Russian).

4. A.V. Kel’manov, S.A. Khamidullin, V.I. Khandeev. “A Fully Polynomial-Time Approximation Scheme for a Sequence 2-Cluster Partitioning Problem,”J. Appl. Indust. Math., 10, No. 2, 209–219 (2016).

81

An approximation algorithm for a problem ofpartitioning a sequence into clusters

Alexander Kel’manov1,2, Sergey Khamidullin1, VladimirKhandeev1,2, Ludmila Mikhailova1


kelm,kham,khandeev,[email protected]

We consider the following strongly NP-hard [1]Problem. Given a sequence Y = (y1, . . . , yN) of points from R

q andsome positive integers Tmin, Tmax, and L. Find nonempty disjoint subsetsM1, . . . ,ML of N = 1, . . . , N such that

L∑

l=1

∑

j∈Ml

‖yj − y(Ml)‖2 +∑

i∈N\M

‖yi‖2 → min,

where M =⋃L

l=1 Ml, and y(Ml) is the centroid of subset yj | j ∈ Ml, underthe following constraints: (1) concatenation of elements of subsets M1, . . . ,ML

is an increasing sequence, provided that the elements of each subset are in as-cending order, (2) the following inequalities for the elements of M = n1, . . . , nMare satisfied: Tmin ≤ nm − nm−1 ≤ Tmax ≤ N, m = 2, . . . ,M ; (the cardinalityof M assumed to be unknown).

In [2], for the case of Problem, when L = 1, a 2-approximation polynomial-time algorithm having O(N2(N + q)) running time was presented. In this workwe present an algorithm that allows to find a 2-approximate solution of Problemin O(LNL+1(N + q)) time, which is polynomial if the number L of clusters isfixed.

The authors were supported by the Russian Foundation for Basic Research(projects no. 15-01-00462, no. 16-31-00186-mol-a, no. 16-07-00168).

References


2. A.V. Kel’manov, S.A. Khamidullin. “An Approximation Polynomial-TimeAlgorithm for a Sequence Bi-Clustering Problem,” Comput. Math. Math.Phys., 55, No. 6, 1068-1076 (2015).

82

An approximation scheme for a balanced 2-clusteringproblem

Alexander Kel’manov1,2, Anna Motkova2



We consider the following strongly NP-hard [1]Problem. Given a set Y = y1, . . . , yN of points from R

q and a positiveinteger M . Find a partition of Y into two non-empty clusters C and Y \ C suchthat

|C|∑

y∈C

‖y − y(C)‖2 + |Y \ C|∑

y∈Y\C

‖y‖2 −→ min ,

where y(C) = 1|C|

∑y∈C

y is the geometric center (centroid) of C, subject to con-

straint |C| =M .

In [2], an exact algorithm for the case of integer components of the inputpoints was presented. If the dimension q of the space is bounded by a constant,then this algorithm has a pseudopolynomial complexity.

In this work we present an approximation algorithm that allows to find a

(1+ε)-approximate solution in O(qN2(√

2qε+1)q) time for a given relative error

ε. If the space dimension is bounded by a constant this algorithm implementsa fully polynomial-time approximation scheme.


References

1. A.V. Kel’manov, A.V. Pyatkin. “NP-Hardness of Some Quadratic Euclidean2-clustering Problems,” Doklady Mathematics., 92, No. 3, 634–637 (2015).

2. A.V. Kel’manov, A.V. Motkova. “Exact pseudopolynomial algorithm for abalanced 2-clustering Problem,” Diskretnyi Analiz i Issledovanie Operatsii(in Russian), 23, No. 3, 21–34 (2016).

83

An approximation scheme for a problem of finding asubsequence

Alexander Kel’manov1,2, Semyon Romanchenko1,2, SergeyKhamidullin1


kelm,rsm,[email protected]

We consider the following strongly NP-hard [1]Problem. Given a sequence Y = (y1, . . . , yN ) of points from R

q, and somepositive integer numbers Tmin, Tmax andM . Find a subset M = n1, . . . , nM ⊆1, . . . , N such that

∑j∈M ‖yj−y(M)‖2 → min, where y(M) = 1

|M|

∑i∈M yi,

under constraints Tmin ≤ nm − nm−1 ≤ Tmax ≤ N, m = 2, . . . ,M, on theelements of (n1, . . . , nM ).

In [2], a 2-approximation polynomial algorithm having O(N2(N + q)) run-ning time was proposed. In [3], for the case of fixed space dimension andinteger input points coordinates, an exact pseudopolynomial algorithm withO(N3(MD)q)-time complexity was presented, where D is the maximum abso-lute coordinate value of the points in the input sequence.

In this work we present a FPTAS for the case of fixed space dimension withO(MN3(1/ε)q/2)-time complexity for an arbitrary relative error ε.


References


2. A.V. Kel’manov, S.M. Romanchenko, S.A. Khamidullin. “Approximationalgorithms for some intractable problems of choosing a vector subsequence,”J. Appl. Indust. Math., 6, No. 4, 443–450 (2012).

3. A.V. Kel’manov, S.M. Romanchenko, S.A. Khamidullin. “Exact pseudopoly-nomial algorithms for some NP-hard problems of searching a vectors sub-sequence,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki (inRussian), 53, No. 1, 143–153 (2013).

84

An algorithm of using the set of equivalence methodfor solving the multicriterial optimization problems

Ruben V. Khachaturov1

1 Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;rv [email protected]

An algorithm of step by step implementation the set of equivalence methodfor solving the multicriterial discrete optimization problems is described. Ad-vantages of the method of finding the set of equivalence for solving this kind ofproblems [1, 2] are shown. The work of this method is illustrated on exampleof a set of key indicators of economic efficiency for generalized commercial en-terprise and the elaborated corresponding mathematical model. In difference tothe classical problem of finding the maximum profit for any business (where theprofit is the only criterion), a multicriterial inverse [3, 4] optimization problemis considered. The solution of the formulated problem in a multidimensionalpseudo-metric space is given.

The optimal sets spacial distributions for each criterion and the set of equiv-alence distribution (violet color) are shown on fig.1 and fig.2.

Fig. 1

85

Fig. 2

References

1. V.R. Khachaturov. A combinatoric-approximation method for the decompo-sition and composition of systems, and finite topological spaces, lattices, andoptimization. USSR Computational Mathematics and Mathematical Physics,Vol. 25 (6), pp. 117–127 (1985).

2. 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).

3. R.V. Khachaturov. Direct and Inverse Problems of Determining the Parame-ters of Multilayer Nanostructures from the Angular Spectrum of the Intensityof Reflected X-rays. Journal of Computational Mathematics and Mathemat-ical Physics, Vol. 49, No. 10, pp. 1781–1788 (2009).

4. R.V. Khachaturov. Direct and Inverse Problems of Studying the Propertiesof Multilayer Nanostructures Based on a Two-Dimensional Model of X-RayReflection and Scattering. Journal of Computational Mathematics and Math-ematical Physics, Vol. 54, No. 6, pp. 984–993 (2014).

86

Approximability of the Euclidean Capacitated VehicleRouting Problem

Michael Khachay1, Roman Dubinin1

1Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia;[email protected], [email protected]

The Capacitated Vehicle Routing Problem is the well known special case ofVehicle Routing Problem [1], which is widely adopted in operations research. Inits simplest setting, the VRP can be treated as the combinatorial optimizationproblem aiming to design the cheapest collection of delivery routes of the givencapacity vehicle from single or multiple dedicated points (or depots) to a set ofcustomers (clients) distributed in a given spatial region.

As the VRP, the CVRP is strongly NP-hard and APX-complete. Almostall known special cases of the CVRP (except the case when q ≤ 2) are alsoNP-hard even in Euclidean spaces of finite dimension.

Most approximation results for CVRP are obtained for the Euclidean plane.One of the first studies of two-dimensional Euclidean CVRP has been due toHaimovich and Rinnooy Kan [2], who presented several heuristics for this prob-lem leading to the first PTAS for q = O(log log n).

In this paper, we extend the results obtained in [2,3] to the case of any fixeddimension d > 1 and fixed number m of depots. Actually, we propose a newEfficient Polynomial Time Approximation Scheme (EPTAS) for the EuclideanCVRP, for which capacity q, the number of depots m, and dimension d > 1 arefixed. The algorithm proposed remains PTAS for the problem with fixed m,d > 1, and q = O(log log n)1/d.

The authors were supported by the Russian Scientific Foundation (projectno. 14-11-00109).

References

1. P. Toth P. and D. Vigo The Vehicle Routing Problem. Society for Industrialand Applied Mathematics, Philadelphia. (2001)

2. M. Haimovich and A. Rinnooy Kan Bounds and Heuristics for CapacitatedRouting Problems. Mathematics of Operations Research, 10 527–542 (1985)

3. M. Khachay and H. Zaytseva Polynomial Time Approximation Scheme forSingle-Depot Euclidean Capacitated Vehicle Routing Problem. LNCS, 9486178–190 (2015)

87

Approximation Shemes for the Generalized TSP inGrid Clusters

Michael Khachay1, Katherine Neznakhina1

1Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia;[email protected], [email protected]

The Generalized Traveling Salesman Problem (GTSP) is a generalizationof the well known Traveling Salesman Problem (TSP). The main difference isthat together with a weighted graph G = (V,E,w) the instance is specified bya partition of the node set V = V1 ∪ . . .∪Vk into disjunctive subsets or clusters.The goal is to find a minimum cost cycle such that each cluster is hit by exactlyone node of this cycle.

We consider a geometric setting of the GTSP, in which a partition is specifiedby cells of the integer 1× 1 grid (on the Euclidean plane). Even in this specialsetting, the GTSP remains intractable enclosing the classic Euclidean TSP onthe plane. Recently [1], it was shown that this problem has (1.5 + 8

√2 + ε)-

approximation algorithm with complexity bound depending on n and k polyno-mially, where k is the number of clusters. We propose three approximation al-gorithms for the Euclidean GTSP on grid clusters. For any fixed k, all proposedalgorithms are PTASs. Time complexities of the first two remain polynomialfor k = O(log n) while the last one is a PTAS when k = n−O(log n). Although,the problem is polynoilly solvable in the case of fixed k, our PTAS’s have rathersmall time complexity (wrt. n) and can be useful in tackling of Big Data.


References

1. B. Bhattacharya et al. “Approximation Algorithms for Generalized MST andTSP in Grid Clusters,” Combinatorial Optimization and Applications: 9thInternational Conference, COCOA 2015, Houston, TX, USA, December 18-20, 2015, Proceedings, LNCS, 9486, 110–125 (2015).

88

Complexity and Approximability of GeometricalPiercing Set Problem for Rectangles Intersecting a

Diagonal Line

Michael Khachay1, Maria Poberiy1

1Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia;[email protected], maschas [email protected]

The classic Hitting Set Problem (HSP) can be stated as follows. For a givenhypergraph G = (V,E), it is required to find a smallest vertex subset H hittingall the hyperedges, i.e. such a subset H that H ∩ e for any e ∈ E. In generalcase, this problem is equivalent to the well known Set Cover problem, it is NP-hard and hardly approximable. Nevertheless, there are many special cases ofthe HSP, which are polynomially solvable or can be approximated well. One ofsuch special cases is called Piercing Set Problem (PSP). In PSP, a hypergraphis given implicitly by a collection of geometric shapes S1, . . . , Sn located on theplane. Basicly, in this case, we are aimed to find the smallest finite subset H ofthe plane hitting all the shapes.

We consider a very special case of PSP, where all the shapes are axis-parallelrectangles intersecting a given straight line (diagonal). It is known that, evenin this special setting, the PSP remains intractable while has a 4-approximatepolynomial time algorithm.

Recently [1], it was shown that, for unit squares, the PSP has an exact algo-rithm with a rather huge but polynomial time-complexity. In the context of thehypothesis P 6= NP , this result seems very intriguing. Indeed, for the majorityof known polynomially solvable problems, there are built exact algorithms withlow-order polynomial complexity bounds.

We extend this result to construct polynomial and pseudo-polynomial exactand approximation algorithms for more several more wide subclasses of the PSP.


References

1. A. Mudgal and S. Pandit Covering, Hitting, Piercing and Packing RectanglesIntersecting an Inclined Line. LNCS, 9486 126–137 (2015)

89

Terminal control problem with fixed ends indynamics: saddle-point technique

Elena Khoroshilova1

1 Lomonosov Moscow State University, Moscow, Russia;[email protected]

In a Hilbert space, the problem of terminal control with linear dynamicsand fixed ends of trajectory is considered. The integral objective functionalhas a quadratic form. In contrast to the traditional approach, the problemof terminal control is interpreted not as an optimization problem, but as asaddle-point problem. The solution to this problem is a saddle point of theLagrange function with components in the form of controls, phase and conjugatetrajectories.

We consider the simplest convex optimal control problem

(x∗(·), u∗(·)) ∈ Argmin

1

2

∫ t1

t0

(〈Q1(t)x(t), x(t)〉+ 〈Q2(t)u(t), u(t)〉)dt ,

d

dtx(t) = D(t)x(t) +B(t)u(t), t0 ≤ t ≤ t1,

x(t0) = x0, x(t1) = x1, x(·) ∈ ACn[t0, t1], u(·) ∈ U .(1)

where Q1(·), Q2(·) are continuous positive semidefinite symmetric n× n, r × r-matrices, D(·), B(·) are n × n, n × r continuous matrices; trajectory x(·) isabsolutely continuous function; set of admissible controls

U = u(·) ∈ Lr2[t0, t1] | ‖u(·)‖L2 ≤ const.

We need to find a control u∗(·) ∈ U such that the corresponding trajectoryx∗(·) connects some starting point x0 with a given point x1 ∈ X1 at the rightend. Problem (1) is a convex programming problem formulated in Hilbert space.We introduce to (1) the Lagrangian

L(ψ(·);x(·), u(·)) = 1

2

∫ t1

t0

(〈Q1(t)x(t), x(t)〉+ 〈Q2(t)u(t), u(t)〉)dt

+

∫ t1

t0

〈ψ(t),D(t)x(t) +B(t)u(t)− d

dtx(t)〉dt

∀ψ(·) ∈ Ψn2 [t0, t1], (x(·), u(·)) ∈ ACn[t0, t1]× U, x(t0) = x0, x(t1) = x1.

By linearization of the Lagrangian and using the saddle-point inequalitiessystem, we get the dual problem to (1). By bringing together main elements

90

of primal and dual problems, we obtain a final system. The solution to thissystem is a saddle point of the Lagrangian, and some of its components formthe desired solution [1],[2].

To solve the problem we build a controlled simple iteration method:

1) predictive half-step

d

dtxk(t) = D(t)xk(t) +B(t)uk(t), xk(t0) = x0,

d

dtψk(t) +DT(t)ψk(t) = −Q1(t)x

k(t), ψk(t1) = 0,

uk(t) = πU (uk(t)− α(Q2(t)u

k(t) +BT(t)ψk(t)));

2) basic half-step

d

dtxk(t) = D(t)xk(t) +B(t)uk(t), xk(t0) = x0,

d

dtψk(t) +DT(t)ψk(t) = −Q1(t)x

k(t), ψk(t1) = 0,

uk+1(t) = πU (uk(t)− α(Q2(t)u

k(t) +BT(t)ψk(t))), k = 0, 1, ...

In linear-convex case, this approach could be interpreted as strengtheningthe Pontryagin maximum principle. It provides the convergence of computingprocess to solution of the problem in all components: the convergence in controlsis weak, the convergence in phase and conjugate trajectories is strong.

The work was carried out with financial support from the Russian Founda-tion for Basic Research (Project No.15–01–06045-a).

References

1. A.S. Antipin, E.V. Khoroshilova. “Saddle point approach to solving problemof optimal control with fixed ends,” Journal of Global Optimization, 65, No. 1,3–17 (2016).

2. A.S. Antipin, E.V. Khoroshilova. “On a boundary-value problem of terminalcontrol with a quadratic quality criterion,” Izvestiya IGU, Matematika, 3,No. 8, 7–28 (2014) [in Russian]

91

Smoothing factor of the frontier transformation inthe DEA models

Vladimir Krivonozhko1, Finn Førsund2, Andrey Lychev3

1 National University of Science and Technology “MISiS”, Moscow, Russia;[email protected]

2 University of Oslo, Oslo, Norway; [email protected] National University of Science and Technology “MISiS”, Moscow, Russia;

[email protected]

At present the DEA models are widely used for performance analysis ofsocio-economic systems. However, some investigators noted that inadequateresults may arise in the DEA models. In our previous papers [1, 2] we discoveredthe reasons of such results. It was also proved that only terminal units give usnecessary and sufficient set of units for smoothing the efficient frontier. Inaddition, a general algorithm was elaborated by our team for smoothing thefrontier. This algorithm was based on using the notion of terminal units. Underthe elaboration of the algorithm we stick to the following principles: a) allefficient units have to stay efficient after the frontier transformation; b) everyinefficient unit will be projected on the efficient part of the frontier.

Moreover, we have to introduce the notion of smoothness of the frontierin order to compare original and transformed frontier. Our computational ex-periments using real-life data sets confirmed that smoothing factor representsadequate measure of the frontier transformation.

The work was carried out with financial support from the Ministry of Ed-ucation and Science of the Russian Federation in the framework of IncreaseCompetitiveness Program of NUST MISiS (Agreement 02.A03.21.004). Thereported study was also partially supported by Russian Foundation for BasicResearch (project no. 14-07-00472).

References

1. V.E. Krivonozhko, F.R. Førsund, A.V. Lychev. “Terminal units in DEA:definition and determination,” Journal of Productivity Analysis, 42, 151–164(2015).

2. V.E. Krivonozhko, F.R. Førsund, A.V. Lychev. “On comparison of differentsets of units used for improving the frontier in DEA models,” Annals ofOperation Research, DOI: 10.1007/s10479-015-1875-8 (2015).

92

Cosmonauts Training Scheduling Problem

Alexander Lazarev1, Nail Khusnullin2, Elena Musatova2, AlekseyPetrov2, Aleksey Gerasimov2, Maxim Kharlamov3, DenisYadrentsev3, Konstantin Ponomarev3, Sergey Bronnikov4

1 National Research University Higher School of Economics, Moscow;Lomonosov Moscow State University, Moscow;

Institute of Physics and Technology State University, Moscow;V.A. Trapeznikov Institute of Control Sciences, Moscow, Russia;

[email protected] V.A. Trapeznikov Institute of Control Science of Russian Academy of

Sciences, Moscow, Russia; [email protected] YU. A. Gagarin Research & Test Cosmonaut Training Center, Star City,

Russia; [email protected];[email protected];[email protected] Rocket and Space Corporation Energia after S.P. Korolev, Korolev, Russia;

[email protected]

Nowadays, in Russia, spaceflight training scheduling is performed manuallyand without using any mathematical approaches. Due to that, fast changesof a training plan will cause a huge workload. We hope that the consideredapproaches and models will lead to reducing these workloads.

Commonly, the cosmonaut training planning is divided into the two stages:the volume planning and the timetabling. In the former one, for each cosmonauta set of tasks is formed depending on requirements of their qualifications andforthcoming on-board experiments complexity conditions. For details, see at[1]. We study the second stage of the problem, i.e. timetabling. There is a setof crews. Each crew consists of a number of cosmonauts. Each cosmonaut hashis own set of training tasks. Dates of the training start and finish are given.The goal is to form a training schedule for each cosmonaut.

From a mathematical point of view, the spaceflight training scheduling canbe considered as a generalization of the resource-constrained project schedulingproblem. This problem is NP-hard. In practice, a planning horizon is about3 years. Each cosmonaut has an individual, aperiodical learning plan. So, theproblem has a very large dimension and is hard to solve.

A mathematical model based on integer linear programming (ILP) is pro-posed. An alternative approach to the problem is to use Constraint Program-ming (CP) solvers. To use this approach, the problem has been reformulatedas a Constraint Satisfaction Problem. The advantage of CP is its possibility toreduce the set of admissible values of variables. The important principle of CPconsists of distinguishing constraint propagation and decision-making search.

93

Table 1: The comparison of ILP and CP

|W | CPLEX ILP CPLEX CPTime,s Var. Constr. Time,s Var. Constr.

4 30.75 52680 60066 0.33 363 27885 559.84 73500 87846 0.44 492 35486 374.63 115200 2022320 0.61 642 43487 346.30 144480 820534 0.64 654 43748 6 657.98 204000 210646 1.32 852 5738

Constraint propagation is a deductive activity which consists in deducing newconstraints from existing constraints. The large number of constraints in ourproblem contributes to high efficiency of CP methods.

The calculations were performed using the solver IBM ILOG CPLEX ver.12.6.2. A distinct advantage of CP is shown in presented Tab. 1. Here |W | is anumber of weeks in planning horizon, Var. is a number of variables, Constr. isa number of constraints in the problem.

Current implementation of the presented model in terms of CP allows usto form schedule with planning horizon equal to 1 year with less than 5 min.The main purpose of this research is to develop mathematical model and findapproaches to solve it in order to implement Planner system in the near future.

References

1. S. Bronnikov, V. Gushchina, A. Lazarev, N. Morozov, A. Sologub and D.Yadrentsev. “Three approaches to solving the problem of cosmonauts train-ing planning”, 7th Multidisciplinary International Conference on Scheduling:Theory and Applications (MISTA 2015), 750–754, 2015.

94

Dynamic programming approaches for single-trackscheduling problem

Alexander Lazarev1, Elena Musatova2, Ilia Tarasov3, Yakov Zinder4

1 National Research University Higher School of Economics, Moscow;Lomonosov Moscow State University, Moscow;

Institute of Physics and Technology State University, Moscow;Institute of Control Sciences, Moscow, Russia;

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

[email protected] Lomonosov Moscow State University, Moscow;Institute of Control Sciences, Moscow, Russia;

[email protected] University of Technology, Sydney, Australia;

[email protected]

We consider a scheduling problem with the single railway track connectingtwo stations. Single railway tracks constitutes the major part of railway trans-port networks in many regions, and are very common in supply chains. Surveyof the railway planning models and methods can be found in the publicationof Lusby et al [1]. Usually problems are considered in terms of scheduling the-ory as job-shop problems, and dynamic programming approach [2] or heuristicmethod [3] is applied.

We investigate the problem where trains travel between two stations, denotethem as Station 1 and Station 2, which are connected by a single track. Eachtrain travels either from Station 1 to Station 2 or from Station 2 to Station 1.The transportation commences at time t = 0. Define the set of all trains atboth stations as N . Two models are considered: the model with the siding on atrack and the model without a siding. For both models we propose the solutionalgorithms based on the dynamic programming method.

The model without a siding is described as follows. There are α differentpossible speeds of trains, where α is a rather small number. For example, wecan divide trains into 3 groups according to the speeds: freight trains, passengertrains, express trains. Each train travels with one of constant α speeds, for eachspeed the train traversing time for the track is given. There must be a minimalsafe distance between two trains simultaneously moving in one direction, so aminimal time interval between the departure of two trains is required, it dependson the speeds of successive trains. In addition, train movement on the track ispossible only in restricted set of time intervals V = t|t ∈ [u1, v1] ∪ [u2, v2] ∪

95

... ∪ [uq , vq ], where q is a number of intervals. We consider the set of objectivefunctions which can be represented with as special general form. For eachtrain some additional parameters depending on objective function are given. Inthe most general case of the problem with α speeds of trains, given possibleinterval of movement V and λ sets of trains with specified departure order the

solution algorithm constructs optimal schedule in O(q2 log qn2α2+2α+1nλ log n)operations, where n = |N | is the number of trains on both stations at the initialmoment. In some cases it is possible to significantly reduce the complexity, ifα = 1, V = t|t ∈ [0,∞), and objective function is maximum lateness, thenthe complexity is O(n2).

In the second model there is the siding on the track, which capacity is onetrain. On the track it is possible to cross two incoming trains only in the siding.All trains have equal constant speed, train traversing times for track segmentsseparated by the siding are given. Minimal time interval is required betweenthe departures of two trains from one station and between the arrival of twotrains to the siding. For each train i ∈ N due date di or priority coefficient wi

are given. Two objective functions which have to be minimized are considered:maximum lateness and the weighted sum of arrival moments. For each objectivefunction the complexity of solution algorithm is O(n2).

The authors were supported by the Russian Foundation for Basic Research(projects 15-07-03141, 15-07-07489) and by The Ministry of Education andScience of Russia, unique identifier – RFMEFI58214X0003.

References

1. R. Lusby, J. Larsen, M. Ehrgott et al. ”Railway Track Allocation: Modelsand Methods”, OR Spectr. Secaucus, NJ, 2011, 33, No. 4, 843–883.

2. E. Gafarov, A. Dolgui, A. Lazarev ”Two-Station Single-Track RailwayScheduling Problem With Trains of Equal Speed”, Comput. and IndustrialEngin., 85, 260–267 (2015).

3. Y. Sotskov, O. Gholami ”Mixed graph model and algorithms for parallel-machine job-shop scheduling problems”, Int. J. of Prod. Res., publ. online,16 (2015). http://dx.doi.org/10.1080/00207543.2015.1075666

96

Application of splines for the evaluation ofinvestment rate dynamics

Valery Lebedev1, Konstantin Lebedev2

1 SUM, Moscow, Russia; [email protected] SRI FRCEC, Moscow, Russia; [email protected]

In the report the results of spline analysis of statistical data used in paper [1]are given. We used statistical data of this work, because the application of mod-ern mathematical methods of analysis cannot only evaluate the dynamics of theparameters of the production function, but also to assess the dynamics of somemacroeconomic indicators in the future. For decision the problem of dynamicsets smoothing the following specific functions applied: polynomial splines (lin-ear, parabolic, cubic) with various defects, functions with piecewise-constantgrowth rate and functions with piecewise-constant values of the coefficient ofelasticity.

Data analysis was accomplished using a computer programme that imple-ments the algorithm for constructing functions with piecewise constant charac-teristics. On the base of spline-analisys it is shown that the production sector ofthe USA from 1899 to 1922 developed unevenly and owing to it the coefficientof elasticity of Cobb-Douglas production function was variable.

Approximation of theoretical dependence of the output-labour ratio on thecapital-labour ratio by functions with piecewise constant elasticity allowed oneto estimate time intervals in which this coefficient can be considered as theconstant. Estimations of the values of the investment rate in 1899–1922 basedon some hypotheses about the dynamics of the coefficient of disposals in thisperiod were obtained. It is shown that statistical data used in paper [1] containinformation that warns of a possible decline in production in US manufacturingin the 5-10 years (after 1922).


References

1. Charles W. Cobb, and Paul H. Douglas. “A Theory of Production”. AmericanEconomic Review, Papers and Proceedings of the Fortieth Annual Meetingof the American Economic Association, 18 (1, Supplement): 139–165 (1928).

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Algorithm of decision of task of smoothing outdynamic rows by functions with piece-permanent

parameters

Konstantin Lebedev1, Tatyana Tyupikova2

1 SRI FRCEC, Moscow, Russia; [email protected] JINR, Dubna, Russia; [email protected]

Actuality of tasks of retrospective analysis is determined by that under-standing of history of development of process and its conformities to law directlyinfluences on quality of prognoses. Therefore, a retrospective analysis must bepreceded to the decision of task of prognostication. For correct authentication ofthe folded tendencies with the purpose of extrapolation, last different methodsand receptions of smoothing out of dynamic rows are used in the future.

Development of this approach for the analysis of production dependenciesis connected, in particular, with the researches, which were carried out underthe supervision of the Prof. Yu.P. Ivanilov in the Computer Center of the Rus-sian Academy of Sciences, by A.P. Abramov, V.A. Bessonov, V.A. Fadeyev,T.I. Gurova, K.H. Zoidov, V.V. Lebedev, K.S. Sviridenko and others in 1985–1996.

In the presented lecture, the algorithm of decision of task of smoothing outof dynamic rows functions is expounded with piece-permanent parameters [1].This algorithm will realize the method of progressing wave for the decision ofsome task of optimal management, to that the task of smoothing out is taken.Substantially, that here in the complement of the varied parameters the valuesof argument enter in the key points of the constructed function. Appropriatecomputer software allows you to build the following specific functions: poly-nomial splines (linear, parabolic, cubic) with various defects, functions withpiecewise-constant growth rate and functions with piecewise-constant values ofthe coefficient of elasticity.


References

1. Yu.P. Ivanilov, V.V. Lebedev. Primenenie splajnov dlja sglazhiva-nija di-namicheskih rjadov. Soobshhenie. VC AN SSSR, Moskva. (1990).

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Relation between Mangasarian-Fromovitz conditionand some other constraint qualifications in nonlinear

programming

Alexey Leschov1, Leonid Minchenko1

1 Belarusian State University of Informatics and Radioelectronics, Minsk,Belarus; [email protected]

Constraint qualifications (CQs) play an important role in nonlinear pro-gramming since they allow to provide the validity of the Kuhn-Tucker necessaryoptimality condition and to construct numerical algorithms for finding optimalpoints.

Consider the set C = y ∈ Rm | hi(y) ≤ 0, i ∈ I, hi(y) = 0, i ∈ I0 of

feasible points in nonlinear programming problem, where I = 1, ..., s, I0 =s + 1, ..., p, hi(y), i = 1, 2, ..., p are continuously differentiable functions fromR

m to R.

The most well-known CQ is the Mangasarian-Fromovitz condition (MFCQ)[1], which holds at a point y ∈ C if Λ0(y) = 0 where

Λ0(y) = λ ∈ Rp |

p∑

i=1

λi∇hi(y) = 0, λi ≥ 0 and λihi(y) = 0 for i ∈ I.

In spite of important place of MFCQ in optimization theory there are prob-lems where MFCQ does not hold though some other CQs can be fulfilled. One ofsuch constraint qualifications is the constant rank condition (CRCQ) by Janin[2]. It is known that MFCQ and CRCQ are independent of each other, i.e.there are examples where MFCQ is satisfied while CRCQ is not and vice versa.The relation between MFCQ and CRCQ was recently studied in the work [3]by Shu Lu where interesting results were obtained. As demonstrated in thiswork, if CRCQ holds at a point y ∈ C then by removing some constraints andtransforming some constraints of inequality type into equalities one can obtaina set which locally does not differ from C but for which MFCQ holds at thispoint.

One of the generalizations of MFCQ is the relaxed Mangasarian-Fromovitzconstraint qualification (RMFCQ) which has been proposed in [4] (see also [5,6]).Later this condition has been introduced in [7] under the name CRSC (constantrank of the subspace component condition).

Let I(y) = i ∈ I |hi(y) = 0 and Ia(y) = i ∈ I(y)|∃λ ∈ Λ0(y)such thatλi >0.

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D e f i n i t i o n 1. RMFCQ holds at a point y0 ∈ C if there exists someneighbourhood V (y0) of this point such that rank∇hi(y), i ∈ I0 ∪ Ia(y0) =rank∇hi(y

0), i ∈ I0 ∪ Ia(y0) for all y ∈ V (y0).

RMFCQ is weaker not only with respect to MFCQ but also it is weaker withrespect to such constraint qualifications as CRCQ and the relaxed constant rankcondition [8], the conditions CPLD and RCPLD [7].

The following theorem generalizes the result by Shu Lu [3] and demonstratesthe relationship between RMFCQ and MFCQ.

Theorem 1. Let the RMFCQ holds at a point y0 ∈ C. Then there existindex subsets Ja ⊂ Ia(y0) and J0 ⊂ I0 such that the set C locally coincides withthe set C# = y ∈ R

m | hi(y) ≤ 0, i ∈ I\Ia(y0), hi(y) = 0, i ∈ J0 ∪ Ja in aneighborhood of y0 and the Mangasarian-Fromovitz consraint qualification holdsat y0 ∈ C#.

References

1. O.L. Mangasarian, S. Fromovitz “The Fritz-John necessary optimality con-ditions in presence of equality and inequality constraints”, J. Math. Anal.and Appl., No. 17, 37–47 (1967).

2. R. Janin “Directional derivative of the marginal function in nonlinear pro-gramming,” Math. Programming Study, No.21, 110–126 (1984).

3. Lu Shu “Implications of the constant rank constraint qualification,” Math.Programming, No.126, 365–392 (2011).

4. L.I. Minchenko, S.M. Stakhovski “To generalizating Mangasarian-Fromovitzregularity condition,” Doklady BSUIR, No.8, 104–109 (2010).

5. A.Y. Kruger, L.I. Minchenko, J.V. Outrata “On relaxing the Mangasarian-Fromovitz constraint qualification,” Positivity, No.18, 171–189 (2014).

6. A.E. Leschov, L.I. Minchenko “Weak regular mathematical programmingproblems,” Izvestija of National Academy of Sciences of Belarus, Ser. Phys.-Math., No.2, 64–70 (2014).

7. R. Andreani, G. Haeser, M.L. Schuverdt, P.J.S.Silva “Two new weak con-straint qualifications and applications,” SIAM J. Optimiz. No.22, 1109–1125(2012).

8. L.I. Minchenko, S.M. Stakhovski “On relaxed constant rank regularity con-dition in mathematical programming,” Optimization, Vol.60, No.4, 429–440(2011).

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Pareto Frontier Visualization in Developing theControl Rules for Angara River Basin

Alexander V. Lotov1, Andrey I. Ryabikov2



Visualization of the Pareto frontier is an efficient tool for decision supportin multiobjective environmental decision problems with many objectives (4 to9 objectives). The Interactive Decision Maps (IDM) technique implements vi-sualization of the Pareto frontier by approximating the Edgeworth-Pareto Hull(EPH), i.e. the maximal set, which Pareto frontier coincides with the Paretofrontier of the feasible objective set, and by subsequent visualization of its Paretofrontier by displaying the decision maps, i.e. overlapped bi-objective slices of theEPH while the value of a third objective is changed [1]. To visualize the Paretofrontier for 4 to 9 objectives, the interactive and animated decision maps areused. Graphic information on the Pareto frontier helps the decision maker orthe negotiators to specify the preferred non-dominated objective point (feasiblegoal point) consciously. Then, the associated decision is provided.

Approximating the EPH is the most complicated step of the IDM technique.In the 1990s, effective methods for polyhedral approximating the convex EPHwere developed. In the 2000s, hybrid methods that integrate classic gradient-based optimization methods and random search techniques with evolutionarymultioblective techniques were developed for non-linear non-convex problemswith up to 9 objectives. In this paper, application of the IDM technique in theframework of the development of control rules for the Angara River reservoirsystem is described.

The research was partially supported by the Program of RAS I.5 P.

References

1. A.V. Lotov, V.A. Bushenkov, G.K.Kamenev. Interactive Decision Maps. Ap-proximation and Visualization of Pareto Frontier, Kluwer, Boston (2004).

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Multi-criteria approach to the analysis of theefficiency of optimization algorithms

Vlasta Malkova1


It’s well-known that the performance of optimization methods heavily de-pends on a selection of method’s parameters values. Moreover usually there istrade-off between various criteria, e.g. running time and obtained optimum. Inthis paper we propose the multi-criteria approach to analyze the impact of themethod parameters on the efficiency of an optimization process.

We evaluate the proposed approach on the problem of minimization of thepotential energy of the 2D-crystal lattice considered in [1]. We study the co-ordinate descent method with the following parameters: Inc — increase of thestep in the case of success and Dec — decrease of the step in the case of failure.

The following approach has been proposed:

1. Choose basic criteria. In our example we choose AverV alue – the averagevalue of the objective function at the found solution obtained after the givennumber of trials and the total running time required to complete these iterations.The first important observation is that the number or trials should be sufficientlylarge to guarantee the stability of the criteria value. Fig.1 shows that for thesmall number of trials the average value vary significantly from one trial toanother and the stabilizes after approximately 150 trials. Further tests werecarried out for a given number of trials N = 150.

2. Building a grid of two parameters optimization algorithm on a given fieldof study with a certain step. On this grid, we have carried out tests and buildthe Pareto front with the rejection of dominated values.

Fig. 1. Average values over trials

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Fig.2 shows the results for AverV alue and T ime criteria, on the left – allpoints, on the right — only non-dominated points. White point represents anideal result (all criteria achieves minimum values). Color of the points indicatestheir proximity to the ideal.

Now it seems clear, that turquoise points are the best ones, but this wereserve to a decision maker.

Fig. 2. Total 42 results obtained, non-dominated 12 ones

The author was supported by the Leading Scientific Schools Grantno. 8860.2016.1.

References

1. Yuri Evtushenko, Mikhail Posypkin. New optimization problems arising inmodeling of 2D-crystal lattices // Proceedings of the NUMTA’2016, Italy,19-25 June, 2016, (in print)

2. S.A. Lurie, M.A. Posypkin, Yu.O. Solyaev “Method of identification of scaleparameters of gradient theory of elasticity on the basis of numerical experi-ments in flat composite structures”, International Journal of Open Informa-tion Technologies ISSN: 2307-8162 vol. 3, no. 6, 1–6 (2015)

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On optimal control of dynamical systems describedby differential inclusions

Olga Masina1, Olga Druzhinina2

1 Yelets State University after I.A. Bunin, Yelets, Russia; [email protected] Institute of Informatics Problems, FRC CSC RAS, Moscow, Russia;

[email protected]

The systems modelled by the differential equations with multiple-valuedright parts are considered in [1–6] and in other works. Some questions of ex-istence and stability of differential inclusions are studied in [1–3]. The sta-bility conditions and stabilization of the systems described by the differentialinclusions containing control in the right parts is obtained in [4] by means ofLyapunov functions method.

In this report the system of movement of object from the start point tothe final point with the an exit in the vertical direction is considered. Thespecified system is described by differential inclusions, and the multiple-valuedcomponents in the right parts of the equations is connected with need of theaccounting of resistance of the rarefied environment.

The problem of optimal control consists in a choice of parameters of draftand values of time so that object motion from the initial point to final withintermediate achievement of height was carried out with the minimal fuel con-sumption.

By means of results of works [1–3] the stability analysis of the nominalmotion determined by criterion of an optimality is performed. The algorithmsof optimal trajectory search are offered for object transition to the purpose inthree-dimensional space. The set of programs realized in Matlab environmentis developed. This set of programs contains of modules for data input and fora graphic illustration of trajectories. The program calculates draft force andvelocities of the control object. We consider also the modifications of modelsdescribed in [5, 6].

The offered algorithms and the set of programs can be used for selectionof optimal parameters of the motion of transport systems in the conditions ofincomplete information.

References

1. A.A. Shestakov. Generalized direct Lyapunov’s method for systems with dis-tributed parameters, URSS, Moscow (2007).

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2. Yu.N. Merenkov. Stability-like properties of differential inclusions, fuzzy andstochastic differential equations, Russian University of Peoples’ Friendship,Moscow (2000).

3. O.N. Masina. ”On the existence of solutions of differential inclusions”, Dif-ferential Equations, 44, No. 6, 872–875 (2008).

4. Bacciotti A. ”Stability and stabilization of discontinuous systems and nons-mooth Lyapynov functions”, Rapporto Interno, 27, 1–21 (1998).

5. O.N. Masina. ”Questions of motion control of transport systems”, Transport:science, technique, control, No. 12, 10–12 (2006).

6. O.V. Druzhinina, O.N. Masina. ”Optimal control for technical systems mod-elled by differential inclusions”, Proceedings of the VI International con-ference on optimization methods and applications ”Optimization and ap-plications” (OPTIMA2015) held in Petrovac, Montenegro, September 27–October 3, 2015. Dorodnicyn Computing Center of RAS, Moscow, 2015,pp. 124–125.

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On a gradient constraint problem for scalarconservation laws

Marin Misur1, Darko Mitrovic2, Andrej Novak3

1 University of Zagreb, Zagreb, Croatia; [email protected] University of Montenegro, Podgorica, Montenegro; [email protected]

3 University of Zagreb, Zagreb, Croatia; [email protected]

We consider a Dirichlet-Neumann boundary problem in a bounded domainfor scalar conservation laws. Using an idea from [1], we propose an informalsolution concept by considering an elliptic approximation to the problem (seeChapter 3 of [2]). We are not yet able to prove existence or uniqueness ofthe solution satisfying the proposed solution concept, but, under appropriateassumptions, we prove that a corresponding weak limit satisfies the consideredequation in the interior of the domain. The basic tool is the compensatedcompactness method. In the case when the flux is continuously differentiablewith respect to all the variables, Remark 1 of [3] implies that a weak limit ofthe elliptic approximation satisfies the Kruzhkov admissibility conditions in theinterior of the domain.

We also provide numerical examples to justify, in the special situation, oneof the limiting assumptions of the theoretical framework.

The research is supported by the bilateral project Multiscale Methods andCalculus of Variations between Croatia and Montenegro; by the Ministry ofScience of Montenegro, project number 01-471; by the University of Zagrebthrough grant PP04/2015; by the Croatian Science Foundation, project number9780 Weak convergence methods and applications (WeConMApp); and by theFP7 project Micro-local defect functional and applications (MiLDeFA) in theframe of the program Marie Curie FP7-PEOPLE-2011-COFUND.

References

1. B. Andreianov, D. Mitrovic. “Entropy conditions for scalar conservation lawswith discontinuous flux revisited”, Annales Inst. Henry Poincare – AnalyseNonlineaire C, 32, 1307–1335 (2015).

2. J. L. Lions, E. Magenes Non-homogeneous Boundary value Problems andApplications I, Springer–Verlag, (1972).

3. E. Yu. Panov. “On weak completeness of the set of entropy solutions to ascalar conservation law,” SIAM J. Math. Anal., 41, 26–36 (2009).

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The estimation of the company’s marketcapitalization based on production models taking into

account the deficit of current assets

Nataliia Obrosova1, Alexander Shananin2

1 MIPT, Dolgoprudny, CC RAS, Moscow, Russia; [email protected] MIPT, Dolgoprudny, CC RAS, Moscow, Russia; [email protected]

Economic downturn in a number of the states of the eurozone (Italy, Spain,Portugal, Greece), and also decrease in growth rates in a number of the coun-tries with economy of the catching-up type (for example Russia and China) isled to decrease in demand in the world markets. The interruptions in salesof products lead to the current assets deficit and requirement of their replen-ishment by credits. Pledge when crediting current assets is capitalization ofthe company which fall leads to increase of interest rates and lowering of thecredit rating of the company. Therefore an important task is development ofthe tool which allows to estimate capitalization of the company on the basis ofindicators of its activities. Such tool can be created based on a class of mathe-matical models of production taking into account current assets deficit which isdeveloped by authors since the end of the 90th years and consistently describesschemes of functioning of production in the conditions of unstable demand atvarious stages of development of the Russian economy [1-2]. The modern ver-sion of model considers restriction of trade infrastructure and formalized in theform of Bellman’s equation for which the solution in an explicit form found [2].The benefit of the model is that it allows calculating the average indicators ofactivities comparable to data of the official reporting of the companies (IFRSstandard). The model allows estimating capitalization of the company depend-ing on indicators of its activities, an external economic environment and size ofa discount of the company’s income that characterizes appeal of the company toinvestors. In terms of model the analysis of influence on dynamics of company’screditworthiness indicator (which is understood as the capitalization relation tocurrent assets), a discount rate of company’s income is carried out that allowsestimating compliance of expectations of the market to real indicators of thecompany. Below are presented the results of estimation of creditworthiness in-dicator for the FCA, Italy and MCC, China in case of the level of a discountrate of income equals to average rate on long-term loans (dashed lines) and thelevel of a discount rate reflecting the expectations of the market which havedeveloped by results of the biddings on the stock exchange (firm lines) (Fig.1).For the FCA company the essential growth of a share price can be explainedwith expansion of a dealer network of the company on the American market due

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to absorption of Daimler Crysler though indicators of activities of the companycorrespond to much lower capitalization. For MCC real capitalization grows ina case, but much more slowly, than share market value that it is possible toexplain with strong long-term state support of the company. The authors weresupported by the RSF, project N16-11-10246.

Fig. 1

The authors were supported by the RSF, project N16-11-10246.

References

1. A.V. Akparova , A.A. Shananin. “The Production Model under Conditionsof Incomplete Credit System and Nonstable Realization of Production.”,Matem. Mod., 16, V.17, N9, 60-76 (2005).

2. N.K. Obrosova, A.A. Shananin. “Production Model in the Conditions of Un-stable Demand Taking into Account the Influence of Trading Infrastructure:Ergodicity and Its Application”, Computational Mathematics and Mathe-matical Physics, 55, N4, 699-723 (2015).

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Identification of a dynamic model of Russianeconomy with two kinds of capital

Nicholas Olenev1,2,3


2 People’s Friendship University of Russia, Moscow, Russia;3 Moscow Institute of Physics and Technology, Moscow, Russia;

Paper [1] proposed an economic model with two kinds of capital - old andnew. It is assumed that the old capital A(t), created during the Soviet time,from 2008 is only eliminated (see [2, 1] for explanations),

A = −µAA(t), A(t0) = A0, (1)

and the new capital B(t) is increased due to investments and is eliminated as aresult of aging.

B = J(t)− µBA(t), B(t0) = B0. (2)

This two kinnds of capital and labour L(t)

L = γL(t), L(t0) = L0. (3)

are three factors of the Cobb-Douglas production function for GDP Y (t).

Y (t) = Y0(A(t)/A0)α(B(t)/B0)

β(L(t)/L0)λ. (4)

For closing our model we can use the same heuristic equations for the GDPcomponents (export E(t), import I(t), investment J(t)) as in [2] and the balanceequation for “consumption” Q(t)) :

E(t) = εY (t)/p(t), (5)

I(t) = ι(Y (t)− p(t)E(t))/q(t), (6)

J(t) = ξ(Y (t) + q(t)I(t))/s(t), (7)

Q(t) = Y (t)− s(t)J(t) + q(t)I(t)− p(t)E(t). (8)

It is proposed an indirect parameter identification method for a Russianeconomic model by statistical time series of macroeconomic indicators of Russia2008-2014. The procedure of identification of the model includes a parallelcomputing on cluster supercomputer as in [3-8].

The author of the work was supported by the Russian Scientific Foundation(project no. 14-11-00432).

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References

1. N.N. Olenev “Identifikatsiya parametrov v modeli ekonomiki Rossii s dvumyavidami kapitala,” in: Informatika, upravlenie i sistemnyiy analiz. Trudyi IVVserossiyskoy konferentsii molodyih uchenyih s mezhdunarodnyim uchastiem,Tom II, Tverskoy gosudarstvennyiy tehnicheskiy universitet, Tver, 2016,pp. 148-155.

2. N.N. Olenev, R.V. Pechenkin, A.M. Chernetsov. Parallelnoe program-mirovanie v MATLAB i ego prilozheniyai, Dorodnicyn Computing Centre,Moscow (2007). DOI: 10.13140/RG.2.1.1766.4481

3. N.N. Olenev, R.V. Pechenkin, A.M. Chernetsov. Parallelnoe pro-grammirovanie v MATLAB i Simulink s prilozheniyami k modelirovaniyuekonomiki, Dorodnicyn Computing Centre, Moscow (2015). DOI:10.13140/RG.2.1.3899.2240

4. V. Dikusar, N. Olenev “Parallel programming in MATLAB for modeling aneconomy,” in: Computer Algebra Systems in Teaching and Research. Vol.5./Eds. Alexander N. Prokopenya, Miroslav Jakubiak.Siedlce University ofNatural Sciences and Humanities, Siedlce, 2015, pp. 63-70.

5. N. Olenev “Economy of Greece: evaluation of real sector,” CEDIMES Scien-tific Seminar at Dorodnicyn Computing Centre, 1, No. 2, 57-72 (2015).

6. N. Olenev “Parallel algorithms of global optimization in identification of aneconomic model,” CEDIMES Scientific Seminar at Dorodnicyn ComputingCentre, 1, No. 1, 45-47 (2015).

7. V.D. Matveenko, N.N. Olenev, A.V. Shatrov “Modeling economic growth ofdifferent countries by means of production functions on the basis of compar-ative analysis of dynamics of interaction of social groups,” Perm UniversityHerald. Economy, No 2(25), 4250 (2015).

8. Kamenev G.K., Olenev N.N. “Study of the Russian Economy’s Identificationand Forecast Stability Using a Ramsey Type Model,” Mathematical Modelsand Computer Simulations, 7, No. 2, 179-189 (2015).

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Improved computing realization of atmosphericgeneral circulation model

Valeriy Parkhomenko

Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;[email protected]

The climate model of the Computing Center of RAS includes an atmo-spheric unit, based on the Atmospheric General Circulation Model (AGCM)with parameterization of a number of subgrid processes, global ocean modeland a model of the evolution of sea ice [1, 2]. Proposed modified method ofAGCM parallelization with calculating the contribution of physics and dynam-ics, respectively, on two groups of processors with the same input data leads toa more efficient and flexible scheme of calculation. Model version with a finerspatial resolution and ocean general circulation model is developed. AGCM isthe software package which simulates many physical processes [3, 4]. There aretwo major program components: AGCM Dynamics block, which calculates thefluid flow described by the primitive equations by finite differences, and AGCMPhysics block which computes the effect of processes not resolved by the modelgrid (such as solar and heat radiative fluxes, internal sub-grid scale adiabaticprocesses, moist and convection processes). The results obtained by AGCMPhysics are supplied to AGCM Dynamics as forcing for the flow and thermo-dynamics calculations . The AGCM code uses a three dimensional staggeredgrid for velocity and thermodynamic variables (temperature, pressure, watervapor mixing ratio, etc.). The AGCM Dynamics itself consists of two maincomponents: a spectral filtering part and the actual finite difference calcula-tions. The filtering operation is needed at each time step in regions close to thepoles to ensure the effective grid size there satisfies the stability requirement forexplicit time difference schemes when a fixed time step is used throughout theentire spherical finite-difference grid. [5]. Processors domain decomposition inthe two-dimensional horizontal plane grid is used in a parallel implementationof the model. This choice is based on the fact that vertical physical processesstrongly link grid points and that the number of grid points in the vertical di-rection is usually small. That makes parallelization less efficient in the verticaldirection. Each subdomain of this grid is a rectangular area that contains allpoints of the grid in the vertical direction. Two types of interprocessor com-munications are mainly in this case [5]. Data exchanges are needed betweenlogically adjacent processors (nodes) in the calculation of finite differences andremote data exchanges are needed, in particular, to carry out the operation ofspectral filtering.

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The analysis shows that the results for the modified scheme give satisfactoryresults and it is possible to use it. Scalar program calculation time takes 33

Modified method advantages are clear for experiments with relatively largeamount of processors, when many interprocessors data exchange exist in Dy-namics block, but Physics block does not have this problem. This effect explainsthe slowing down of original method calculations [7]. Number of processors, dis-tributed for Dynamics block in the modified method is the same for all exper-iments, and number of processors, distributed for Physics block increase withthe more detailed description of physical processes in the model experiments.

The author was supported by the Russian Foundation for Basic Research(projects no. 14-07-00037, 14-01-00308, 16-01-00466).

References

1. Parkhomenko V.P. Sea Ice Cover Sensitivity analysis in Global ClimateModel. Research activities in atmospheric and oceanic modelling. WorldMeteorological Organization Geneva Switzerland in 2003, V. 33, p. 7.19 -7.20.

2. Parkhomenko V.P. Numerical experiments on global hydrodynamic model toassess the sensitivity and climate sustainability. Vestnik MGTU im. Bau-mana. Issue Mathematical Modelling, p. 134-145 (2012) (in Russian).

3. Moiseev N.N., Aleksandrov V.V., Tarko A.M. Man and the Biosphere. Ex-perience in systems analysis and experiments with models. Moscow: Nauka,1985. 272 p. (In Russian)

4. Ghan S.J., Lingaas J.W., Schlesinger M.E., Mobley R.L., Gates W.L. Adocumentation of the OSU two-level atmospheric general circulation model.Rep. No. 35, Climatic Research Institute. Oregon State University, Corvallis,395 p. (1982).

5. Parkhomenko V.P. V.P. Parkhomenko The implementation of a global cli-mate model on a multiprocessor computer cluster type. Parallel ComputingTechnologies (PaVT’2009): Proceedings of the International Scientific Con-ference (Nizhny Novgorod, March 30 - April 3, 2009). - Chelyabinsk Univ.South Ural State University, 2009. p.644-652. (In Russian)

6. Parkhomenko V. P., Tran Van Lang. Improved computing performance andload balancing of atmospheric general circulation model. Journal of ComputerScience and Cybernetics. V. 29. N 2. P. 138 148. (2013).

7. Voevodin V.V., Voevodin Vl.V. Parallel computing. St. Petersburg.: BHV-Petersburg, 2002. 600 p. (In Russian).

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Ellipsoidal estimation of the attraction domain foraffine systems with constrained control resource

Alexander Pesterev1

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

We consider a class of single-input affine control systems that can be repre-sented in the canonical form [1]

x1 = x2, . . . , xn−1 = xn, xn = f0(x) + f1(x)u, (1)

where x ∈ Dx ⊆ Rn, f0(x) and f1(x) 6= 0 are continuous functions, and u is ascalar control. Applying feedback u = −(σ(x)+f0(x))/f1(x), where σ(x) = cTx,cT = [c1, . . . , cn], ci > 0, we obtain a linear closed-loop system. While ci’s canalways be selected such that the latter system is globally asymptotically stable,the original system (1) cannot be linearized in the entire domain because of aconstrained control resource. Hence, the origin is only locally stable, and wearrive at the problem of finding (an estimate of) the attraction domain of anonlinear closed-loop system.

Assuming that the control u is subject to the constraint |u| ≤ u and definingthe feedback in the entire domain Dx to be

u(x) = −satu[(σ(x) + f0(x))/f1(x)], (2)

where satu[·] is the saturation function, we set the following problem: given asystem of form (1) closed by feedback (2), find the “best” ellipsoidal estimateΩ(P ) = x : xTPx ≤ 1, P > 0 ⊆ Dx of the attraction domain of the zerosolution. Under the “best” estimate, we mean, e.g., the ellipsoid for which thetrace of the corresponding matrix P is minimal.

With regard to (2), the closed-loop system can be written as

x1 = x2, . . . , xn−1 = xn, xn = −Φ(σ, x). (3)

Along with (3), we introduce the linear nonstationary system x = Aβ(t)x:

x1 = x2, . . . , xn−1 = xn, xn = −β(t)σ(x), (4)

System (4) is said to be absolutely stable in a sector [β0, 1] if its zero solution isasymptotically stable for any functions β(t) satisfying the condition 0 < β0 ≤β(t) ≤ 1 [2]. If, ∀x ∈ Rn, Φ(σ, x) lay in the same sector, i.e.,

0 < β0 ≤ Φ(σ, x)/σ ≤ 1, (5)

113

then system (3) would also be absolutely stable [2]. If (5) does not hold in Rn

(like in our case), for an estimate of the attraction domain, one can take aninvariant ellipsoid of system (4) in which condition (5) holds [3]. The followingtheorem reduces finding of such an ellipsoid to solving a system of linear matrixinequalities (LMIs).

Theorem. Let Ω(P ) ⊆ Dx and minx∈Ω(P ) U(x) ≥ U0 > 0, where U(x) =|f1(x)|u− |f0(x)|. Let P > 0 be a solution of the LMI system

PA1 + AT1 P ≤ 0, PAβ0 + AT

β0P ≤ 0, P ≥ ccTβ2

0/U20 , (6)

for some 0 < β0 ≤ 1, where A1 and Aβ0 are constant matrices obtained fromAβ(t) by the substitution of 1 and β0 for β(t). Then, Ω(P ) is an invariantellipsoid belonging to the attraction domain of system (3).

To ensure the inclusion Ω(P ) ⊆ Dx and fulfillment of the inequalityminx∈Ω(P ) U(x) ≥ U0, it is proposed to approximate Dx by a family of do-mains Π(U0) ⊆ Dx, 0 < U0 < |f1(0)|u− |f0(0)|, where Π(U0) is a domain witha boundary composed of first- and second-order surfaces satisfying the condi-tion minx∈Π(U0) U(x) = U0. The inscribing of an ellipsoid into such a domainreduces to solving an LMI system li(P ) ≤ 0, i = 1, . . . ,m.

Then, having solved the latter LMI system jointly with (6) by means of, say,Matlab procedure mincx, which finds P with the minimal trace, one gets thebest ellipsoidal estimate for given β0 and U0 with the corresponding performanceindex F(U0, β0) = traceP (U0, β0). By minimizing this function of two variables,one gets the desired ellipsoidal estimate.

This work was supported by the Presidium of RAS (Program I.31. Section“Actual problems of robotics”).

References

1. F.F. Zhevnin and A.P. Krishchenko. “Controllability of nonlinear systemsand synthesis of control algorithms,” Dokl. Akad. Nauk SSSR, 258, 4, 805–809, 1981.

2. E.S. Pyatnitskii. “Absolute stability of nonstationary nonlinear systems,” Au-tomation and Remote Control, 31, No. 1, 5–15 (1970).

3. A.V. Pesterev. “Estimation of attraction domains for canonical forms of affinesystems with constrained control resource,” Automation and Remote Control(in press), (2016).

114

Synchronization, self-organization andself-optimization in nonlinear dynamical systems

Lev Petrov1

1 Plekhanov Russian University of Economics, Moscow, Russia NationalResearch University Higher School of Economics, Moscow, Russia;

[email protected]

Synchronization effects in linear and quasi-linear dynamical systems are wellstudied [1]. Similar effects in essentially nonlinear systems can be detected innumerical experiments with dynamical models [2]. The synchronization andself-organization effects study in strongly nonlinear dynamical systems withdeterministic chaos is of special interest [3]. In this study effective research toolof the solutions becomes an interactive search mode [4].

Synchronization effects in multidimensional nonlinear dynamic systems inmechanics and physics associated with the phenomenon of self-organization inmore complex systems - the economy and society. The phenomenon of self-organization can be interpreted as the natural spontaneous dynamical systemtransition to optimal condition.

We discuss an alternative approach when the objective function to deter-mine the system parameters is the product of deterministic chaos. Cases high-lighted when the self-organization can be associated with a self-optimization.The objective function in this case is formed by natural way.

References

1. I.I. Blehman Synchronization in nature and technology, Nauka, Moscow(1981).

2. Lev Petrov “Nonlinear effects in economic dynamic models” NonlinearAnalysil, 71, 2366–2371 (2009).

3. Lev Petrov Economy dynamic analysis methods,Infra-M, Moscow (2010).

4. Lev Petrov “Interactive computational search strategy of periodic solutionsin essentially nonlinear dynamics,” in: Interdisciplinary Topics in AppliedMathematics, Modeling and Computational Science, Springer InternationalPublishing ,Switzerland, 2015, pp. 355–360.

115

ARIMA-GARCH models of RTS and MICEX futuresdynamics

Aleksandr Petrovykh1

1 Lomonosov Moscow State University, Moscow, Russian Federation;[email protected]

In Russia, the most liquid futures contracts on stock indices are futures onRTS and MICEX indices. To assess the market and credit risk of derivativesportfolio the Monte Carlo method based on the simulation of multiple futurescenarios of initial uncertainty factors is commonly used. On the basis of simu-lated future factor values for each scenario corresponding derivatives prices arecalculated.

In the basic theory of index futures pricing based on the concept of perfectmarket in the absence of arbitrage opportunities the ”Cost of Carry” modelis used to assess the price of the stock index futures. The implementation ofthe ”Cost of Carry” model for the futures on RTS and MICEX indices showedthat the difference between the calculated and actual futures prices may besignificant.

This report describes causes of price differences, presents the applicationresults of existing stock index futures pricing models to futures on RTS andMICEX indices, and offers a number of new index futures pricing models basedon a modification of ”Cost of Carry” model, that can be applied within theframework of the Monte Carlo method for the derivatives portfolio risk assess-ment. Proposed models are based on ARIMA-GARCH processes with differentexplanatory factors and conditional distribution functions. The comparativestatistical analysis of proposed models based on Akaike, Schwarz and Hannan-Quinn information criteria is presented.

References

1. J.C. Hull. Options, futures and other derivative securities, Prentice Hall, NewJersey (2011).

116

On alternative duality and symmetric lexicographicalcorrection of improper linear programs

Leonid D. Popov1, Vladimir D. Skarin2

Krasovskii Institute of Mathematics and Mechanics UB RAS, Ekaterinburg,Russia;


Consider the dual pair of linear programs

max(c, x) : Ax ≤ b, x ≥ 0, (1)

min(b, y) : AT y ≥ c, y ≥ 0, (2)

where A = (aij)m×n, c ∈ Rn and b ∈ R

m are given, x ∈ Rn is the vector of

primal variables, y ∈ Rm is the vector of dual variables, (·, ·) means the scalar

product. Assume that programs (1)–(2) may be improper [1] (i.e. one of theseprograms or both of them may be infeasible). Suppose that initial constraintssystems are divided onto series of subsystems arranged by their permissibilityfor relaxation. It means that initial data in (1)–(2) is parted onto blocks likethis:

c

x

A b y

∼

c0 c1 . . . cn0

x0 x1 . . . xn0

B0 B1 . . . Bn0 b y

∼

c

x

A0 b0 y0A1 b1 y1...

......

Am0 bm0 ym0

To determine the generalize (relaxed) solution for (1)–(2) write out the dualpair of symmetric relaxed programs

max(c−∆c, x) : Asx ≤ bs +∆bs (s = 0, . . . ,m0), x ≥ 0, (3)

min(b+∆b, y) : BTs y ≥ cs −∆cs (s = 0, . . . , n0), y ≥ 0, (4)

where ∆c = [∆c0, . . . ,∆cn0 ] ∈ Rn and ∆b = [∆b0, . . . ,∆bm0 ] ∈ R

m are theparameters of relaxation. The vector ∆c = [∆c0, . . . ,∆cn0 ] is called an optimallexicographical relaxation of objective vector in (3)–(4) iff

∆c0 = argmin‖∆c0‖ : BT0 y ≥ c0 −∆c0, y ≥ 0, . . . ,

∆cn0 = argmin‖∆cn0‖ : BTn0y ≥ cn0 −∆cn0 ,

BTs y ≥ cs −∆cs (s = 0, . . . , n0 − 1), y ≥ 0.

117

Analogously, the vector ∆b = [∆b0, . . . ,∆bm0 ] is called an optimal lexicograph-ical relaxation of rhs-vector in (3)–(4) iff

∆b0 = argmin‖∆b0‖ : A0x ≤ b0 +∆b0, x ≥ 0, . . . ,

∆bm0 = argmin‖∆bm0‖ : Am0x ≤ bm0 +∆bm0 ,

Asx ≤ bs +∆bs (s = 0, . . . ,m0 − 1), x ≥ 0.

In this report it is described the connection between these vectors and thesolutions of the dual pair of convex programs

P : maxx≥0

Φσ(x), Φσ(x) = (c, x)−n0∑

s=1

βs2‖xs‖2 −

m0∑

s=1

1

2αs‖(Asx− bs)

+‖2,

D : miny≥0

Ψσ(y), Ψσ(y) = (b, y) +

n0∑

s=1

1

2βs‖(BT

s y − cs)+‖2 +

m0∑

s=1

αs

2‖ys‖2,

generated as maxmin and minmax of the regularized Lagrangian

Lσ(x, y) = (c, x)− (y,Ax− b)−n0∑

s=1

βs2‖xs‖2 +

m0∑

s=1

αs

2‖ys‖2.

This work extends previous results [2] of authors onto more general case.

This work was partially supported by Russian Foundation for Basic Research(project no. 16-07-00266).

References

1. I.I. Eremin, V.D. Mazurov, and N.N. Astaf ’ev. Improper Problems of Linearand Convex Programming, Nauka, Moscow (1983) (in Russian).

2. L.D. Popov, V.D. Skarin. Lexicographical regularization and alternative du-ality schemas for improper linear programming programs, Trudy Inst. Math-emat. i Mechan. UrO RAN, 21:3, 279–291 (2015) (in Russian).

118

Singular optimization and p-regularity theory

Agnieszka Prusinska1, Alexey Tret’yakov1,2,3

1 Siedlce University, Siedlce, Poland; [email protected] Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;

3 System Research Institute, Polish Academy of Sciences, Warsaw, Poland;[email protected]

Consider the nonlinear programming (NLP) problem with inequality con-straints

minϕ(x) subject to g1(x) ≤ 0, . . . , gm(x) ≤ 0, (1)

where ϕ and gj , j = 1, . . . ,m are smooth functions from Rn to R

The Lagrangian for the problem (1) is defined as

L(x, λ) = ϕ(x) +m∑

j=1

λjgj(x), where λ = (λ1, . . . , λm)T is a vector of Lagrange

multipliers. The Kuhn-Tucker (KK) conditions of optimality are satisfied at x∗

with some λ∗ ∈ R if

L′x(x

∗, λ∗) = ϕ′(x∗) +

m∑

j=1

λ∗jg

′j(x

∗) = 0 (2)

λ∗jgj(x

∗) = 0, λ∗j ≥ 0, gj(x

∗) ≤ 0, j = 1, . . . ,m. (3)

There are various methods for solving the NLP problem based on the KK sys-tem. However, these approaches usually require some regularity constraintsqualification, second-order sufficient conditions (SOSC) and strict complemen-tarity conditions (SCC). Moreover we have to take into account feasibility ofconstraints, i.e. gj(x

∗) ≤ 0, j = 1, . . . ,m. For the quadratic convergence of theNewton method for the system (2)–(3) it is necessary SOSC and SCC as well.There are numerous of NLP problems for which these assumptions are failed.

Our approach is based on the construction of p-regularity theory (see [1,2])and on transforming the inequality constraints into equalities. Namely, by in-troducing slack variables s1, . . . , sm we get the equivalent equality constrainedproblem

minϕ(x) subject to g1(x) + s21 = 0, . . . , gm(x) + s2m = 0. (5)

The necessary optimality conditions for this problem are

F (x, s, λ) =(L′(x, s, λ)

)T=

(L′

x(x, s, λ), L′s(x, s, λ), L

′λ(x, s, λ)

)T= 0, (6)

119

where L(x, s, λ) = ϕ(x) +m∑

j=1

λj(gj(x) + s2).

To solve the system (6) by Newton method we require that matrixF ′(x∗, s∗, λ∗) is not singular. If the SCC does not hold at x∗, i.e. there ex-ists an index j such that λ∗

j = 0 and s∗j = 0 then F ′(x∗, s∗, λ∗) is singular. Ourgoal is to apply the p-factor method for p = 2 to solve this singular optimizationproblem. The main scheme of p-factor method (p = 2) for the system (6) is

zk+1 = zk −F ′(zk) + P2F

′′(zk)h−1

·(F (xk) + P2F

′(zk)h), (7)

where z = (x, s, λ)T and P2 is a matrix of the orthoprojection onto (ImF ′(z∗))⊥.D e f i n i t i o n. The mapping F : Rn+2m → R

n+2m is called 2-regular atz∗ along an element h ∈ R

n+2m, if a matrix F ′(z∗)+P2F′′(z∗)h is nonsingular.

For our optimization problem the following result will be hold.Theorem 1. Let f, g ∈ C3(Rn). Assume that for x∗ ∈ R

n, there exists aLagrange multiplier λ∗ ∈ R

m satisfying (2)-(3) and SOSC holds. Then Lagrangeoptimality system (6) is either regular or 2-regular atz∗ = (x∗, s∗, λ∗)T along some vector h ∈ R

n+2m.Moreover for the scheme (7) the following convergence rate is valid: ‖zk+1−

z∗‖ ≤ C‖zk − z∗‖2, k = 01, . . . .

Research of the second author is supported by the Russian Foundation forBasic Research (project no. 14-07-00805) and by Leading Scientific Schools(grant no. 8860.2016.1) and by the Presidium Programme (I.33P RAC).

References

1. O.A. Brezhneva, Yu.G. Evtushenko, A.A. Tret’yakov. “The 2-factor-methodwith modifid Lagrangian function for degenerate optimization problems,”Doklady Math., 73, 439–442 (2006).

2. A.A. Tret’yakov, J.E.Marsden. “Factor-analysis of nonlinear mappings: p-regularity theory,” Comun. Pure Appl. Anal., 2, 425–445 (2003).

120

Optimization actuators movements of robotic systemwith perturbation effect

L.A. Rybak1, Y.A. Mamaev1, D.I. Malyshev1

1 Belgorod State Technological University of V.G. Shukhov, Belgorod, Russia;rl [email protected]

Consider the problem of synthesis of optimal control of a robotic system withan electromechanical actuator and transmission screw- nut, which is describedin the following state-space equations:

x = kem·r1

Jr+m·r1·r2· Ia + y

Ia = −RL· Ia − kem

r1·L· x+ 1

L· u+ kem

r1·L· y (1)

or in vector-matrix form X = A·X+B·u+A·G·Y, whereX =[x1 x2 x3

]T—

the state vector;The generalized scheme of vibration-proof robot system is shown in figure

1, where 1 - the object of vibration protection; 2 - base; 3, 4 - accelerometers onthe subject and the base; 5 - the gauge of the relative movement; 6 - control; 7- the electric motor; 8 - power amplifier; 9 - spindle; 10 - a nut. Shown in thefigure 2 block ”Object of control”, describes a system of equations.

Figure 1: Driving robotic sys-tem with an electromechani-cal drive.

Figure 2: Generalized sys-tem of vibration protectionscheme with optimal control.

We define the structure of the optimal discrete controller in the form ofa matrix relation u

[i]= −F · X

[i], where F =

[f1 f2 f3

]— Matrix of

the feedback factor in the variable state. In fact, the optimal controller acts

121

as a feedback role of the state of the system. To solve the optimal control

problem of synthesis will form a vector F controlled variables F =

[z1z2

]=

[x1

x2

]= D ·X, where D =

[1 0 00 1 0

]—matrix of communications coordinates

state and controlled variables. As a criterion of quality will take the integral ofquadratic forms of controlled variables and manipulated species:

J =

∞∫

0

(q1 · z21(t) + q2 · z22(t) + r · u2(t))dt→ min (2)

where q1,q2,r —The weighting coefficients of the squares of the two controlledvariables and manipulated respectively, that characterize the contribution ofeach term in the criterion function. The weights should be chosen empirically,based on the results of mathematical modeling. For discrete systems [1] criterionfunction (3) in the form:

J =∞∑

i=0

(XT [i]DT ·Q ·D ·X

[i]+ r · u

[i]) → min (3)

where Q =

[q1 00 q2

]—matrix of weighting factors under controlled variables.

Thus, we have formulated the task of management is to minimize the amplitudeof the movement speed of the object (and hence the amplitude of its accelera-tion) and to limit the level of movement of the object relative to the base, thuslimiting the available-resource management. Feedback gain matrix is given byF = (r+BT

∆ ·P·B∆)−1 ·BT∆ ·P·A∆, where P —positive definite square auxiliary

matrix dimension . The matrix P satisfies the discrete Riccati equation [2]

P = DT ·Q ·D+AT∆ ·P ·A∆ −AT

∆ ·P ·B∆(r+BT∆ ·P ·B∆)−1 ·BT

∆ ·P ·A∆

This equation has a solution, if the pair (A,B) is completely controllableand has a unique positive definite solution in the form of a symmetric matrixP , and such optimum closed system is definitely stable, since all eigenvaluesof the matrix (A − B · F)in modulus less than one. Solving Riccati equationnumerically and highlighting positive definite matrix F, we find control u

[i]in

form u[i]= −f1 · x

[i]− f2 · z

[i]− f3 · I

[i], where the matrix of the feedback

122

factor has the form F =[553, 7196 587, 3993 0, 9638

]. The modeling of the

robotic system when disturbance action on the part of the base with a frequencyf = 1Hz. At the same time suppressing the disturbing factor of the vibrationexposure in amplitude was Ax/Ay = 0, 08. Relative movement of the object islimited and can be held at an acceptable level. The level of the steady relativemovement can be reduced by increasing the weight q2 in the criterion function(3). However, it should be borne in mind that in this case will increase theamplitude of the acceleration on the object.

This work was supported by the Russian Science Foundation (agreement16-19-00148).

References

1. Magergut V.Z. Approaches to the construction of discrete models of con-tinuous technological processes for the synthesis of control automatic /Magergut V.Z., Ignatenko V.A., Bazhanov A.G., Shaptala V.G. // Her-ald of the Belgorod State Technological University named after V.G.Shukhov.2013.num.2.p.100-102.

2. Polyak, B.T., Shcherbakov P.S. Superstable linear control system I: Analysis// Automation and Remote Control.2002.num.8.p.37-53.

123

New Parallel Multi-Memetic MEC-based Algorithmfor Loosely Coupled Systems

Maxim Sakharov1, Anatoly Karpenko2

1 Bauman MSTU, Moscow, Russia; [email protected] Bauman MSTU, Moscow, Russia; [email protected]

This work presents a new parallel adaptive population-based algorithm forsolving global unconstrained optimization problems, which utilizes a multi-memetic approach. The algorithm is based on the concept of Mind Evolu-tionary Computation (MEC) [1]. In turn, multi-memetic approach implies thehybridization of an arbitrary global optimization technique with several localoptimization methods (memes).

The proposed method can be easily considered as a multi-population al-gorithm since it works with a certain number of sub-populations S = (si, i ∈[1 : K]) which is determined by the number of avaliable computational coresK. Each sub-population then evolves independently from one another whichleads to minimization of communicational expenses between nodes. Such afeature makes it promising to use this algorithm on grid-systems composed ofpersonal computers. In addition, each sub-population implements its own evo-lution strategy as well as different values of algorithm’s free parameters.

At the initialization stage, a certain number of initial points (individuals) isgenerated in accordance with LPτ sequence [2]. This set of individuals is dividedinto sub-populations by the value of objective function. Individuals within aparticular sub-population then evolves following modified memetic MEC algo-rithm. Two memes were used in this work, namely the Nelder-Mead methodand the Gauss-Seidel method [3].

In addition, the algorithm implies an adaptive strategy in order to a of solv-ing global optimization problems based on the population Mind EvolutionaryComputation (MEC) [1]. The algorithm can be parallelized easily and can runon loosely-coupled computing systems with distributed memory. This distinc-tive feature makes it promising to use on grid-systems composed of personalcomputers (desktop grids) [2]. Since desktop grids are widely spread nowadaysdue to relatively low cost the development of algorithms for this kind of systemsis an important task.

MEC simulates some aspects of human behavior in the society. Each in-dividual in this algorithm is regarded as an intelligent agent, operating in thearbitrary group of people. In order to achieve a high score, an individual has tostudy under the most successful individuals of his group. The same principle isused in the intergroup competition.

124

Canonical algorithm was modified by authors to prevent the preliminaryconvergence. Operation of decomposition of the search region was added at thevery begining of the algorithm. Besides, hybridization of MEC with memeticalgorithm was proposed.

A memetic algorithm was introduced as a metaheuristic method that isbased on a certain combination of any population algorithm and one or morelocal optimization methods (memes). Overall scheme of the memetic algorithmis as follows. Group initialization S = (si, i ∈ [1 : |S|]), where si is the agent of apopulation and |s| is the total number of agents, takes place at the first iterationof the algorithm (t = 0). Then all operators of the population algorithm areapplied to every agent of the population S. On the basis of new positions ofevery agent si, local search is carried out, that is memes are launched fromthese positions. After the local search the quality of new positions of agentsis estimated. Then the iteration counter is incremented (t = t + 1) and thetermination conditions are tested.

Presented scheme of the memetic algorithm is flexible enough and allowsone to modify it in many different ways and, in particular, to create a multi-memetic algorithm. In this case the swarm of available memes is created M =(mj , j ∈ [1 : |M |]); the most efficient of them then improves the current positionof the agent si. The key problems when developing multi-memetic algorithmsare selecting the most efficient meme and choosing best strategies to controlmemes application.

Described algorithm was implemented by the authors as well as all memeswith a use of Wolfram Language. The application has a unified interface whichenables a usage of various local optimization methods and separate module forcontrolling different adaptive strategies.

Performance investigation of the proposed parallel multi-memetic algorithmand its implementation was carried out using eight-dimensional benchmark func-tions [4]. Numerical experiments were based on the concept of multi-start withk = 100 launches. Averaged best obtained value of objective function F , stan-dard deviation σ and mean number of iterations λ were used as criteria for per-formance evaluation. Obtained results demostrainted that proposed algorithmand its software implementation are capable of determining a global minimumof multiextremal functions with a high probability.

Further studies will be devoted to the development of new adaptive strate-gies which would also take user’s expertise into account.

125

References

1. M. Sakharov, A. Karpenko. “Performance Investigation of Mind EvolutionaryComputation Algorithm and Some of Its Modifications”, Proceedings of theFirst International Scientific Conference Intelligent Information Technologiesfor Industry (IITI16): Volume 1, Springer International Publishing, 2016, pp.475-486.

2. I.M. Sobol. “Distribution of points in a cube and approximate evaluation ofintegrals”, in: USSR Comput. Maths. Math. Phys. Volume 7, 1967, pp.86-112.

3. J.A. Nelder, R. Mead. “A Simplex Method for Function Minimization”, Com-puter Journal. Volume 7, No. 04, 1965, pp. 308-313.

4. J.J. Liang, B.Y. Qu, P.N. Suganthan. “Problem Definitions and EvaluationCriteria for the CEC 2014 Special Session and Competition on Single Ob-jective Real-Parameter Numerical Optimization”, Technical Report 201311.Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou,China; Technical Report. Nanyang Technological University, Singapore, 2013.32 p.

126

A Continuous Model of Rhythmical Production

V.P. Savelyev1,2, A.A. Shamin2

1,2 Nizhny Novgorod State University, Nizhniy Novgorod, Russia;[email protected]

1 Nizhny Novgorod State Engineering-Economic University, Knyaginino,Russia; [email protected]

In this paper, the model of rhythmical production [1–3] in discrete time isdeveloped for the case of continuous time.

Let p(t), u(t), x(t), 0 ≤ t ≤ T , be continuous function representing the rateof supply of raw materials for production of some product, piece-wise continuousfunction representing the rate of elaborating of the raw materials and continuousfunction representing the quantity of raw materials in the stock of the volumeV accordingly. The balance of raw materials in the stock may be described bythe equation

x(t) = x(0) +

∫ t

0

p(τ )dτ −∫ t

0

u(τ )dτ, 0 ≤ t ≤ T. (1)

The total amount P (t) = x(0)+∫ t

0p(τ )dτ of raw materials to be elaborated

and the total amount v(t) =∫ t

0u(τ )dτ of elaborated raw materials are to satisfy

the inequality0 ≤ x(t) = P (t)− v(t) ≤ V. (2)

A piece-wise continuous function u(t), 0 ≤ t ≤ T , satisfying the inequality (2)and the equality

∫ T

0

u(τ )dτ = Q, Q ∈ [P (T )− V, P (T )], (3)

is called to be feasible control. A feasible control minimizing the functional

J [u] =

∫ T

0

f(u(τ ))dτ, (4)

the function f(u) of the class C1 being strictly convex, is called to be optimalcontrol.

Theorem 1. If 0 ≤ P (t)− QTt ≤ V , 0 ≤ t ≤ T , then u0(t) ≡ Q

T, 0 ≤ t ≤ T ,

is optimal control.

Theorem 2. Let u0(t) and x0(t), 0 ≤ t ≤ T , be optimal control and optimaltrajectory accordingly. If 0 < x0(t) < V at some closed interval [t1, t2] thenu0(t) ≡ const for t ∈ [t1, t2].

127

Theorem 3. Let u0(t), 0 ≤ t ≤ T , be optimal control having a discontinuityat a point t∗:

a) if u0(t∗ − 0) < u0(t∗ + 0), then x0(t∗) = 0;

b) if u0(t∗ − 0) > u0(t∗ + 0), then x0(t∗) = V .

Corollary 1. Let u0(t), x0(t), 0 ≤ t ≤ T , be optimal control and optimaltrajectory accordingly, and the equality

x0(t) = 0,(x0(t) = V

), (5)

is held at some point t ∈ (0, T ). Then u0(t− 0) = u0(t+ 0) and x0(t) = 0.

Corollary 2. If 0 < x(0) < V , and 0 < x(T ) < V , the optimal trajectoryconsists of some number of arcs 0 < x0(t) < V , τi < t < ti, i = 1, m, τ1 = 0,tm = T , corresponding to constant values of optimal control u0(t) ≡ ci, linkedby pieces of bounds x0(t) ≡ V or x0(t) ≡ 0, determined at the closed intervals[ti, τi+1], i = 1, m− 1. All arcs are tangent to according pieces of bounds atthe points τi and ti. Moreover, if x0(t) ≡ V (x0(t) ≡ 0) at the closed interval[ti, τi+1], then ci ≥ ci+1 (ci ≤ ci+1).

References

1. V.P. Savelyev, V.N. Fokina “On optimal Rhythmical Production,” in: Fi-nal Program and Abstracts of the Sixth SIAM Conference on Optimization,Atlanta, 1999, pp. 137.

2. V.P. Savelyev. “Optimization of work balancing and minimizing changes ofproduction intensity,” Vestnik NNGU, No. 4, 115–119 (2007).

3. V.P. Savelyev. “A problem of rhythmical production,” in: VI InternationalConference on Optimization Methods and Applications (OPTIMA-2015),Petrovac, Montenegro, September 2015. PROCEEDINGS, Moscow, 2015,pp. 156–157.

128

Iterative Equilibrium Searching in Piecewise LinearExchange Model

V.I. Shmyrev1

1 Sobolev Institute of Mathematics, Siberian Branch, Russian Academy ofScience, Novosibirsk, Russia; [email protected]

The exchange model with piecewise linear separable concave utility func-tions is considered. This consideration extends the author’s original approachto the equilibrium problem in a linear exchange model and its variations. Theconceptual base of this approach is the scheme of polyhedral complementarity.It has no analogs and made it possible to obtain the finite algorithms for somevariations of the exchange model. Especially simple algorithms arise for linearexchange model with fixed budgets (Fisher’s model). This is due to monotonic-ity property inherent in the models and potentiality of arising mappings. Thealgorithms can be interpreted as a procedure similar to the simplex-method oflinear programming.

It is natural to study applicability of the approach for more general models.The considered piecewise linear version of the model is reduced to a specialexchange model with upper bounds on variables and the modified conditions ofthe goods’ balances.

For such a model the monotonicity property is violated. But it remains, ifupper bounds are substituted by financial limits on purchases. A version of thepolyhedral complementarity algorithm for this type of models was developed byauthor earlier.

This is the idea of proposed iterative algorithm for initial problem. On eachstep of the process the approximating model is formed. The equilibrium pricevector of this model is used to get the next approximation. This approach hasbeen proved earlier for general case of linear exchange model.

References

2. V. I. Shmyrev. “Polyhedral complementarity and equilibrium problem in lin-ear exchange models”, Far East Journal Of Applied Mathematics 82, 2 , 67–85(2013).

2. V. I. Shmyrev. About the searching in the linear exchange model with fixedbutgets and additional restrictions on purchases (in Russian). Optimization45(62), 66–86 (1989).

3. V. I. Shmyrev., N. V. Shmyreva. An iterative algorithm for searching anequilibrium in the linear exchange model. Sib.Adv.Math. 6, 1, 87–104 (1996).

129

The solution of an applied problem with mixedconstraints

Alexander Skiba1


We consider a dynamic model of functioning for a group of gas depositswith interacting wells [1]. The optimum control problem is put and solved overa finite horizon with mixed constraints.

Problem 1. About maximizing cumulative production for a group of gas de-posits,

We want to maximize the functional

n∑

i=1

T∫

0

qi(t)Ni(t) dt (1)

in differential relations

qi = − q0iV 0i

qi(t)Ni(t), i = 1, 2, . . . , n, (2)

with initial conditionsq0i > 0, i = 1, 2, . . . , n (3)

and under restrictions on controls

0 6 Ni 6 Ni, i = 1, 2, . . . , n, (4)

n∑

i=1

qi(t)Ni(t) 6 Q, (5)

Controls Ni(i = 1, 2, . . . , n) belong to the set of measurable functions. Theright end of the phase trajectory is free.

The variables have the following designations: Ni is a fund of well productionat the i-th deposit; Ni is the upper limit of the well production fund; qi is theaverage well production rate at i-th deposit; Vi denotes recoverable reservesin the i-th deposit; Q is the total ”shelf” of deposits. The group consists of n

deposits. It is assumed that the valuesq0iV 0i

Ni are different and all gas deposits

are sorted by valuesq0iV 0i

Ni in ascending order. These values are interpreted as

the percentage of gas extraction from the deposits.

130

The existence of an optimal control solution follows, for example, from thetheorem, given in [2, §4.2]. To solve problem 1 we use the proposition formulatedby the Nobel laureate in Economics K. Arrow[3].

The optimal exploitation policy for deposits is as follows. First, at timet = 0 the first deposit is chosen for the exploitation. If the capacity of the firstdeposit is not enough to make gas production at the level of the total value of”shelf” deposits, i.e. q01N1 6 Q, then the second deposit is put into exploitation.If the capacity of the first deposit is enough to carry out the extraction of gasat the level of the total value of ”shelf” deposits, i.e. q01N1 > Q, then the firstdeposit development continues with production at the level of Q. In this case,some of these wells are disabled. The average production rate of wells at firstdeposit reduces over time. For the same reasons gas production at other activewells can fall.

New unused wells are included in the exploitation to compensate for the fallin gas production. At time t = τ1 all wells from the first deposit are included intogas production. The second deposit enters into the exploitation to compensatefor the loss of total gas production. All wells from the first and second depositsare included in the gas production at time t = τ2. The compensation mechanismcontinues till the moment t = Tmax. All wells in all deposits are involved in thegas production at time t > Tmax.

References

1. R.D. Margulov, V.R. Khachaturov, V.A. Fedoseev. System analysis in long-term planning of gas production, Nedra, Moscow (1991).

2. E.B. Lee, L. Markus. Foundations of optimal control theory, Nauka, Moscow(1972).

3. K.J. Arrow. ”Applications of control theory to economic growth,” Mathemat-ics of the Decision Sciences, No. 2, 85-119 (1968).

131

Exact penalization and global optimality conditionsin nonconvex optimization

Alexander Strekalovsky1

1 Matrosov Institute for System Dynamics and Control Theory of SB RAS,Irkutsk, Russia; [email protected]

Consider the optimization problem:

(P) :f0(x) ↓ min

x, x ∈ S ⊂ IRn,

fi(x) ≤ 0, i ∈ I := 1, . . . ,m,

(1)

where all fi = gi(x) − hi(x), i ∈ I ∪ 0 with smooth convex functionsgi(·), hi(·), gi, hi : IR

n → IR, i ∈ I ∪ 0.Let introduce the l∞-penalty function [1]–[3]

W (x) := max0, f1(x), . . . , fm(x) = max0, fi(x), i ∈ I. (2)

Further, consider the penalized problem as follows (σ > 0)

(Pσ) : Θσ(x) := f0(x) + σW (x) ↓ minx, x ∈ S. (3)

As well-known [1]–[3], if z ∈ Sol(Pσ), and z ∈ D := x ∈ S : fi(x) ≤ 0,i ∈ I, then z ∈ Sol(P). In addition, if z ∈ Sol(P), then under supplemen-tary conditions [1]–[3] for some σ∗ > 0, σ∗ ≥‖ λz ‖1 (where λz is the KKT-multiplier corresponding to z), the inclusion z ∈ Sol(Pσ) holds. Moreover,Sol(P) = Sol(Pσ), so that Problems (P) and (Pσ) turn out to be equivalent∀σ ≥ σ∗.

It can be readily seen that the penalized function Θσ(·) is a d.c. one, sincethe functions fi(·), i ∈ I ∪ 0, are as such. Actually, since σ > 0,

Θσ(x) = Gσ(x)−Hσ(x), Hσ(x) := h0(x) + σ∑i∈I

hi(x),

Gσ(x) := Θσ(x) +Hσ(x) =

= g0(x) + σmax

m∑i=1

hi(x);maxi∈I

[gi(x) +∑j 6=i

hi(x)]

,

(4)

it is clear that Gσ(·) and Hσ(·) are convex functions. For z ∈ S denote ζ :=Θσ(z).

Theorem 1. It z ∈ Sol(Pσ), then

∀(y, β) : Hσ(y) = β − ζ, (5)

132

the following inequality holds

Gσ(x)− β ≥ 〈∇h0(y) + σ∑

i∈I

∇hi(y), x− y〉 ∀x ∈ S. (6)

So, Theorem 1 reduces nonconvex (d.c.) Problem (Pσ) to a solving thefamily of convex linearized problems of the form(PσL(y)) : Gσ(x)− 〈∇Hσ(y), x〉 ↓ min

x, x ∈ S,

depending on the parameters (y, β) fulfilling the equation (5).If for such a pair (y, β) and some u ∈ S the inequality (6) is violated, i.e.

Gσ(u) < β + 〈∇Hσ(y), u − y〉,then due to convexity of Hσ(·) we obtain withthe help of (5) that Gσ(u) < β +Hσ(u) −Hσ(y) = Hσ(u) + ζ. It implies thatΘσ(u) = Gσ(u) − Hσ(u) < ζ := Θσ(z), so that u ∈ S is better that z, i.e.z /∈ (Pσ).

It means that Global Optimality Conditions (5), (6) of Theorem 1 possessesthe constructive (algorithmic) property allowing to construct local and globalsearch methods for solving Problem (Pσ) [4].

In particular, they enable us to escape a local pit of (Pσ) and to reacha global solution. Moreover, the Global Optimality Conditions to Problem(Pσ) provide for KKT-conditions at z for the original Problem (P). So, theGlobal Optimality Conditions (5), (6) of Theorem 1 is connected with classicaloptimization theory [1]–[3].

This research is supported by the Russian Science Foundation (grant 15-11-20015).

References

1. J.-F. Bonnans, J.C. Gilbert, C. Lemarechal, C.A. Sagastizabal. NumericalOptimization: Theoretical and Practical Aspects, 2nd ed., Springer-Verlag,Berlin ( 2006).

2. A.F. Izmailov, M.V. Solodov. Newton-Type Methods for Optimization andVariational Problems, Springer, New York (2014).

3. J.V. Burke. “An exact penalization viewpoint of constrained optimization,”SIAM J. Control and Optimization, 29, No. 4, 968–998 (1991).

4. A.S. Strekalovsky. “ On Solving Optimization Problems with Hidden Non-convex Structures,” in: Optimization in Science and Engineering, Springer,New York, 2014, pp. 465–502.

133

Protection and safety for people optimization inemergency situations with radiation leakage

Nikolay Tikhomirov

Plekhanov Russian University of Economics, Moscow, [email protected]

In cases of accidents at the nuclear power plants and terror attacks with“dirty bombs” International Commission on Radiological Protection (ICRP)recommends the principles of cost optimization for the residual dose of ionizingradiation as a corroboration of measures on liquidation of radioactive contami-nation effects and radiological protection. The optimal value of this dose is one,which balances the harm from exposure and intervention costs [1, pp.86-87].

An optimization of intervention costs requires a corroboration of the crite-rion that is adequate to the nature of this goal. The possible criteria are quitediverse. The most common criteria are are the following:

Maximization of intervention benefits:

B = ”value.of.averted.harm”− ”cost.of.intervention” (1)

Minimization of intervention costs:

Z = ”loss.from.residual.dose” + ”cost.of.intervention” (2)

Minimization of intervention costs rate:

K =”cost.of.intervention”

”value.of.averted.harm”(3)

Maximization of specific cost-effectiveness:

V =”value.of.averted.harm”

”cost.of.intervention”(4)

In cases of certain characteristics of the criterion function from the residualdose, its optimum estimate can be determined by means of the methods ofunconditional optimization:

dcriteriondresidual.dose

= 0 (5)

In some studies, these criteria are assumed to be equivalent [2]. However,there is an obvious identity only for the 1st and 2d criteria and the 3rd and the4th ones.

134

The solutions of the optimization problem for such intervention measuresas a decontamination and a decontamination with temporary resettlement ofthe population indicate that the optimum estimates of the residual doses arehigher with the last two criteria, compared with the first ones. This resultindicates that the criteria focused on improving the specific cost-effectivenessunderestimate its social effects, measured as the health and life loss due toradiation and the results of their decrease.

This paper was prepared with financial support of the grant provided by“Russian Foundation for Humanities” (RFH), project number 15-02-00412 “Riskassessment and risk management related to loss of life and health of the popu-lation in emergency situations with radiation leakage”.

References

1. The 2007 Recommendations of the International Commission on RadiologicalProtection. ICRR Publication 103//Annals of the ICRR: Elsilvier. 2007.v. 37. 24. 343 p.

2. Crik M.J. Derived International Levels for Invoking Countermeasures in theManagement of Contaminated Agricultural Enviromments/Division of Nu-clear Safety, IAEA. Vienna, 2009. 23 p.

135

Healthcare expenditure optimization subject tohealth burden in Russian regions

Tatiana Tikhomirova, Valeriya Gordeeva

Plekhanov Russian University of Economics, Moscow, Russiat [email protected]

Increase in the health care effectiveness in Russia is directly related to therational distribution of financial, material and human resources between the re-gions, that is impossible without reliable estimations of the health burden dueto morbidity and premature mortality. However, health loss estimations in theregions of Russia by means of developed methods is limited due to insufficientcompleteness of statistical data on morbidity and mortality in Russian regions.The study was dedicated to the development of procedure for determining thehealth burden from premature mortality and regional healthcare systems ef-fectiveness evaluation, based on existing morbidity and mortality databases inRussian regions.

According to the proposed approach, the years of life lost due to prematuremortality have been determined as the difference between the threshold levelof life expectancy at birth and the average age of death from specific causes.For example, in the global burden of disease studies (indicators DALY, DALE,HALE, etc.) threshold level of life expectancy is 92 years [2]. In order to ensurecomparability of estimates over time and between different areas, there werecalculated age-standardized indicators of the average age of death [1].

There were calculated estimates of years of life lost due to premature mortal-ity from the basic reasons (certain infectious and parasitic diseases, neoplasms,diseases of the circulatory system, diseases of the respiratory system, diseasesof the digestive system) in Russian regions for the period 2006 – 2013 years.

Based on a comparison of the obtained health loss indicators from prematuremortality and healthcare expenditure there were corroborated conclusions inrelation to the cost-effectiveness of resource allocations in regional healthcaresystems. According to the proposed approach, Russian regions were dividedinto four areas, corresponding to different levels of costs and health burden:high loss and high costs (area I), low loss and high costs (area II), low loss andlow costs (area III), high loss and low costs (IV). It was used a median as athreshold for separating high and low levels of costs and health burden. In theregions situated in an area of high rates of losses and high costs (area I) activitiesaimed at strengthening the population health cannot provide social welfare. Inregions with low losses and high costs (area II) it is advisable to carry out anadditional analysis of the situation in order to clarify the reasons for low losses.

136

These regions can be considered as a potential source of resources for the otherregions of the Russian Federation, if the results of additional analysis confirm asurplus in financing, that does not have a significant impact on public health. Inregions with high losses and low level of investment in health there is a problemof insufficient funding (area IV).

The proposed approach to the definition of loss due to premature mortalitycan be used for evaluation of the current effectiveness of regional healthcaresystems, as well as for monitoring the economic impact of measures aimed atreducing the health burden in regions of the country.

This study was supported by the grant provided by “Russian Foundationfor Humanitie” (RFH), project number 15-02-00412 “Risk assessment and riskmanagement related to loss of life and health of the population in emergencysituations with radiation leakage”.

References

1. Tikhomirova T.M., Gordeeva V.I. On the evaluation of the risks of cancerincidence and mortality according to age-specific structure of the population.- Voprosy Onkologii 2014. 60 (5). p. 571-577.

2. WHO methods and data sources for global causes of death 2000-2012:Global Health Estimates Technical Paper WHO/HIS/HIS/GHE/2014.7. Re-source: http://www.who.int/healthinfo/statistics/GlobalCOD method.pdf(05.02.2016).

137

Algorithms with performance guarantees for somehard assignment and rooting problems

Oxana Yu. Tsidulko1,2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia;2 Novosibirsk State University, Novosibirsk, Russia.

[email protected]

The classical linear assignment problem (AP) can be formulated as follows:given an input cost matrix C = (cij), the problem is to find a permutationπ ∈ Sn of size n, such that

n∑

i=1

ciπ(i) → min (max).

This is a polynomially solvable problem. It is known that a permutation can berepresented as a decomposition into disjoint cycles. Modifying the AP by addingdifferent conditions on the number of cycles in the permutation, one may obtaindifferent hard routing problems. An obvious example, the additional conditionfor the permutation π to be a cycle leads to the classical NP-hard Travelingsalesman problem (TSP). The assignment problem with given number m ofcycles in the permutation π is known as the NP-hard m-Cycles Cover Problem.

Another natural modification of the linear AP is a multi-index assignmentproblem (MIAP). It is applied in the practical problems of communication,logistics, production, and economics. Researchers study variations of MIAPwith additional constraints: axial or planar statements, kinds of permutations,etc. The requirements for the permutations in MIAPS to be single cycles alsogive different hard routing problems.

In this work the following results are presented.

1. In 2000 A. I.Serdyukov introduced the m-index axial assignment problemon single-cycle permutations of size n. It turned out that for m > 2 and even nthe problem does not have any feasible solutions. So the question of solvabilityof this problem in case of odd n was studied in [3]. It was shown that forevery 2 < m ≤ 8, there is a number nm such that the axial m-index problem issolvable for odd n > nm.

2. Another studied problem is the 3-index m-layered planar assignmentproblem on single-cycle permutations of size n or the m-Peripatetic salesmenproblem (m-PSP). This is an NP-hard problem. In [1, 2] the approximation al-gorithms for the m-PSP are created for the m-PSP with different and identicalweight functions on random inputs. The conditions under which the algorithms

138

are asymptotically optimal are obtained in the cases when the input data (ele-ments of m×n×n matrix) are independent and identically distributed randomreals with uniform distribution on [an, bn], 0 < an < bn, shifted exponentialdistribution on [an,∞), 0 < an, or any distribution function that dominates theuniform or shifted exponential distribution functions.

3. In [4] the approach to solving the mentioned above m-Cycles Cover Prob-lem was developed. The approach transforms an approximation TSP algorithminto an approximation m-CCP algorithm. In this report the extended range ofsuccessful transformations with proven performance guarantees for the obtainedsolutions is presented.

The author is supported by the RSCF grant 16-11-10041.

References

1. E.Kh. Gimadi, A.M. Istomin, I.A. Rykov, O.Yu. Tsidulko. ”Probabilisticanalysis of an approximation algorithm for the m-peripatetic salesman prob-lem on random instances unbounded from above.” Proceedings of the SteklovInstitute of Mathematics 289, No 1, 77–87 (2015).

2. E.Kh. Gimadi, A.M. Istomin, O.Yu. Tsidulko. ”On Asymptotically Opti-mal Approach to the m-Peripatetic Salesman Problem on Random Inputs.”Proceedings LNCS, submitted (2016).

3. O. Yu. Tsidulko. ”On solvability of the axial 8-index assignment problemon one-cycle permutations”, Journal of Applied and Industrial Mathematics8(1), 115–126 (2014)

4. E. Kh. Gimadi, I. A. Rykov. ”Asymptotically optimal approach to the ap-proximate solution of several problems of covering a graph by nonadjacentcycles” (in Russian), Trudy Inst. Mat. i Mekh. UrO RAN 21(3), 89–99(2015).

139

Automatic Differentiation in Python

Andrei Turkin1

1 Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia;National Research University of Electronic Technology, Zelenograd, Russia;

[email protected]

Nowadays, in many applications one should deal with the problem of exactderivative calculation. One of the possible solutions is to use automatic dif-ferentiation, which may be implemented in Python by means of the followingautomatic differentiation tools, which are the object of this research: PyADOL-C, PyCppAD, CasADi, Computation Graph Toolkit (CGT), Theano, or AD.These tools are analyzed to highlight the advantages of using each of them.

Because of the fact that in many applications the computation of derivativesis the main task, it is important to use an efficient tool, which has maximal run-time speed. Moreover, it should have such precision of gradient calculation thatis almost equal to machine one. Therefore, performance and precision each ofthe tools should be assessed. For these purposes a cluster optimization problem,which can be described as follows, was chosen. We are looking for a geometricalstructure of the cluster with identical atoms, the interaction between which isdescribed by Lennard-Jones pair potentials:

v(ρ) = ρ−12 − 2ρ−6, (1)

by using the following objective function:

E(x) =∑

i<j

v(ρ(xi − xj)), (2)

The cluster database with 1610 different configurations was used to testeach of these tools as follows. First, the average time of gradient calculationwas found to assess performance. Second, by using predefined clusters, the normof the gradient was calculated for each of these clusters to assess precision. Itshould be noted here that despite the usage of the dataset with optimal clusterconfigurations, the precise optimum was found by using optimize.minimize

function of the SciPy package with the L-BFGS method. It was necessary dueto rounding that took place when the configurations were stored to the dataset.

It was shown that the PyADOL-C and PyCppAD tools have much betterperformance for big clusters than the other ones. The precision of these two toolswas assessed by calculating the difference between gradient norms, which wereobtained at different optimal configurations. It was concluded that PyCppAD

140

has the best performance among others, while having almost the same precisionas the second-best performing tool - PyADOL-C.

References

1. Y. G. Evtushenko. Optimization and fast automatic differentiation. Dorod-nicyn Computing Center of Russian Academy of Sciences, 2013. [Online].Available: http://www.ccas.ru/personal/evtush/p/198.pdf.

2. A. Griewank, A. Walther. Evaluating Derivatives: Principles and Techniquesof Algorithmic Differentiation. Society for Industrial Mathematics, 2nd edi-tion, November 2008.

5. S. F. Walter. PyADOL-C: a python module to differentiate complex algo-rithms written in python. Available: www.github.com/b45ch1/pyadolc/

6. B. M. Bell, S.F. Walter. Pycppad: Python algorithmic differentiation usingcppad. Available: http://www.seanet.com/ bradbell/pycppad/pycppad.htm

7. J. Andersson. A General-Purpose Software Framework for Dynamic Opti-mization, October 2013.

8. B. Stadie, Z. Xie, P. Moritz, J. Schulman, J. Ho. Computational graphtoolkit: a library for evaluation and differentiation of functions of multidi-mensional arrays.

9. F. Bastien, P. Lamblin, R. Pascanu, J. Bergstra, I. J. Goodfellow, A. Berg-eron, N. Bouchard, and Y. Bengio. Theano: new features and speed im-provements. Deep Learning and Unsupervised Feature Learning NIPS 2012Workshop, 2012.

11. A. D. Lee. AD: python package for first- and second-order automatic differ-entiation. Available: http://pythonhosted.org/ad

12. M.A. Posypkin. Searching for minimum energy molecular cluster: Methodsand distributed software infrastructure for numerical solution of the problem.Vestnik of Lobachevsky University of Nizhni Novgorod, (1):210 219, 2010.

13. D.J Wales and J.P.K. Doye. Global optimization by basin-hopping andthe lowest energy structures of Lennard-Jones clusters containing up to 110atoms. The Journal of Physical Chemistry A, 101(28):51115116, 1997.

141

Optimization of transmission system for energymarket

Alexander Vasin1

1 Lomonosov Moscow State University, Moscow, Russia; [email protected]

Markets of energy resources play an important role in economies of manycountries. We consider a problem of social welfare optimization for such marketswith account of production costs, consumers’ utilities and costs of trasmissioncapasities’ increments. In general the problem of transport system optimizationis NP-hard (see Guisewite, Pardalos, 1990). Below we determine conditions forsubmodularity and for supermodularity of the social welfare function on the setof transmitting lines. These properties provide a possibility to apply the knownefficient optimization methods (see Khachaturov, 1989).Consider a market including several local markets and a network transmissionsystem. Let N denote the set of nodes and L ⊆ N × N be the set of edges.Every node i ∈ N corresponds to a perfectly competitive market. Demandfunction Di (p) and supply function Si (p) characterize respectively consumersand producers in the market. The demand function relates to the consumptionutility: Ui (q) =

∫ q

0D−1

i (v) dv. The supply function Si (p) determines the op-timal production volume: Si (p) = Arg maxv(pv − ci(v)) , where ci(v) is theminimal production cost of volume v at node i. The total profit of producersat node i under price p is Pri (p) =

∫ p

0Si(p)dp. Every line (i, j) ∈ L is char-

acterized by initial transmission capacity Q0ij , unit transmission cost eijt , cost

function of the transmission capacity increment including fixed costs eijf and

variable costs eijv (Qij , Q0ij), e

ijv is a monotonous convex function of increment

(Qij − Q0ij). Let qij denote the flow from market i to market j, qij = −qji.

Denote Z (i) the set of nodes connected with node i. Under any fixed flows ofthe good −→q = (qij , (i, j) ∈ L) and production volumes −→v = (vi, i ∈ N), thetotal social welfare for the network market is

W (−→q ,−→v ) =∑

i∈N

[Ui

vi +

∑

l∈Z(i)

qli

− ci (vi)]−

∑

(i,j)∈L, i<j

Eij (qij),

Eij (qij) =

eijf + eijv

(|qij | −Q0

ij

)+ eijt |qij | , if |qij | > Q0

ij ,

eijt |qij | , if |qij | ≤ Q0ij .

The welfare optimization problem under consideration is

max−→q ,−→v

W (−→q ,−→v ) (1)

142

For any L ⊆ L, consider a problem (1) with fixed set L of expanded lines.

Let W (L) denote the maximal welfare in the latter problem. Then problem

(1) reduces to maxL⊆L W (L). Below we also consider problem (1) without

construction costs and under constraint: |qij | ≤ Qij , (i, j) ∈ L. Let pi(−→Q), i ∈

N , denote the equilibrium prices corresponding to the solution of this problem.A function w(L), L ⊆ L, is submodular (resp. supermodular) on L, if for

any L1, L2 ⊆ L w(L1) + w(L2) > (6) w(L1 + L2) + w(L1 ∩ L2).Theorem 1. For a chain-type market with n nodes, let the initial prices

pi(−→Q0), i = 1, .., n, monotonously decrease in i. Then, for any

−→Q ≥ −→

Q0,

pi(−→Q) ≥ pi+1(

−→Q), i = 1, .., n − 1, and function W (L) is supermodular. The

complexity of search for the optimal set L∗

under−→Q0 = 0 does not exceed

(n−1)n2

.Consider a star-type market where N = 0, 1, .., n, L = (0, i), i = 1, .., n,

pi(−→Q0) < p0(

−→Q0) for i ∈ I1 = 2, .., m, pi(

−→Q0) > p0(

−→Q0) for i ∈ I2 = m +

1, .., n. For M ⊆ L, let (−→Q0||−→Q∞

M ) denote vector−→Q such that Ql = Q0

l forl /∈M , Ql = ∞ for l ∈M .

Theorem 2. Let ∀i ∈ I1 pi(−→Q0||−→Q∞

I1) < p0(

−→Q0||−→Q∞

I1) and

∀i ∈ I2 pi(−→Q0||−→Q∞

I2) > p0(

−→Q0||−→Q∞

I2). Then the social welfare function W (L1 ∪

L2) is submodular in L1 ⊆ I1 under a fixed set L2 ⊆ I2, and is also submodularin L2 ⊆ I2 under a fixed set L1 ⊆ I1. Besides that, for any L1, l ∈ I1 \ L1, the

welfare function increment W (l∪L1, L2)−W (L1, L2) monotonously increases in

the set L2, and for any L2, l ∈ I2 \L2, the increment W (L1, l∪L2)−W (L1, L2)monotonously increases in the set L1.

The authors were supported by the Russian Foundation for Basic Research(project no. 16-01-00353/16).

143

The computational technique for approximation ofnonlinear functional differential equations of

pointwise type

Tatiana Zarodnyuk1, Armen Beklaryan2, Fedor Belousov3

1 Matrosov Institute for System Dynamics and Control Theory of SB RAS,Irkutsk, Russia; [email protected]

2 National Research University Higher School of Economics, Moscow, Russia;[email protected]

3 Central Economics and Mathematics Institute of the Russian Academy ofSciences, Moscow, Russia; [email protected]

Functional differential equations of pointwise type (FDEPT) attracted theattention of researchers is not only an interesting mathematical object, butalso as a tool for investigating of applied problems in various scientific fields.An important application is the possibility of a reduction of infinite ordinarydifferential equations ODE to small dimension FDEPT infinite ODE (see., eg.,[1]—[2]). However, this approach can not be efficiently applied in practice due tothe underdevelopment of numerical methods for solving initial boundary valueproblems for such systems.

It is considered the system

Fi(x(g(t)), x(g(t))) = 0, i = 1, n, t ∈ [t0, t1],where Fi : Rn ×Rn → R1.

The values of the phase variables derivatives are determined beyond thebasic range t ∈ [tN , tK ], tN ≤ t0, tK ≥ t1.

xLi = hL

i (t), t ∈ [tN , t0] and xRi = hR

i (t), t ∈ [t1, tK ], i = 1, n.

The boundary conditions are given by functionals

Kj(g(x(τj)), g(x(τj))) = 0, j ∈ [1, l], τj ∈ [t0, t1].

A more detailed description of the problem statement can be found in thispaper [3].

To approximate the original problem on the fixed grid of nodes at the timethe trajectories are approached by using spline functions. The coefficients ofthese functions are selected by finding the minimum of residual functional

I(x(t)) =n∑

i=1

vNi

∫ t0

tN

(xi(g(t))− hLi (t))

2dt+

144

n∑

i=1

vNi

∫ tK

t1

(xi(g(t))− hRi (t))

2dt+

l∑

j=1

K2j (g(x(τj)), g(x(τj))) +K0(g(x(τ0)), g(x(τ0))) → min,

where vNi , i = 1, n and vKj , j = 1, l are the weighting coefficients for residualsand boundary conditions.

It has not been possible the using of well-known types of splines (”natural”,with fixed boundary conditions, Akima splines, etc.) for achieving a good es-timation accuracy of the trajectories. Therefore we developed the algorithm ofbuilding ”controlled” splines: it is selected the appropriate values for the bound-ary conditions - the second derivatives at the endpoints of the grid and searchedspline coefficients at the same time. The proposed algorithms are implementedwithin the software OPTCON-F [3].

The results of computational experiments demonstrated the effectiveness ofthe proposed technology for research of considerable problems.


References

1. L.A. Beklaryan. Introduction to the theory of functional differential equa-

tions. The group approach, Factorial Press, Moscow (2007).

2. G.V. Demidenko, S.V. Uspenskii. Equations and systems that are not allowed

for the highest derivative, Science Book, Novosibirsk (1988).

3. T.S. Zarodnyuk, A.S. Anikin, E.A. Finkelshtein, A.L. Beklaryan, F.A. Be-

lousov “The technology for solving the boundary value problems for nonlinear

systems of functional differential equations of pointwise type,” Modern tech-

nologies. System analysis. Modelling, 1, No. 49, 19–26 (2016).

145

Algorithms for global optimization based onCurvilinear Search

Tatiana Zarodnyuk1, Alexander Bugerya2, Fedor Khandarov3

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

Irkutsk, Russia; [email protected] Keldysh Institute of Applied Mathematics RAS, Moscow, Russia;

[email protected]

3 Scientific and educational innovation center of system research and

automation, Institute of Mathematics and Computer Science, BSU, Ulan-Ude,


Applied optimization problems are as a rule multiextreme. Existing ap-

proaches are resultative for limited class of problems or they impose strict re-

strictions on the task setting. Currently, it is actual the development of new

effective approaches to solve the problems of global optimization.

We propose to employ curvilinear search technique for solving multiextremal

optimization problems. This approach based on using of random initial and

auxiliary points for the constructing of curves covering the range of permissible

values.

The search of the smallest value of objective function is carried out along

these lines on each iteration [1]. We can choose the method of generating the

auxiliary points and selection of one-dimensional search algorithm for construct-

ing the different modification of the algorithm.

The proposed technique was applied for solving some problems from dif-

ferent areas, for example, problems from Mongolian industry (it is research

competitive companies which share the bread market of the city Ulaanbataar)

and nanophysics (it is simulated the quantum logic operations in system of

quantum dots).

146


(project no. 15-37-20265).

References

1. A.Yu. Gornov, T.S. Zarodnyuk: Optimization, Simulation, and Control,

chap.206 Tunneling Algorithm for Solving Nonconvex Optimal Control Prob-

lems, pp. 289207 299. Springer New York (2013).

147

Determining parameters of hydrological model

Elena Zasukhina1, Sergey Zasukhin2

1 Computing Center RAS, Moscow, Russia;; [email protected] Moscow Institute of Physics and Technology, 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. Then a wa-

ter transfer can be described by one-dimensional nonlinear parabolic equation.

Following initial-boundary value problem is considered:

∂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 of diffu-

sion; K(θ) is the hydraulic conductivity; R(t) is precipitation; E(t) is evapora-

tion; θmin and θmax are minimal and maximal values of humidity respectively.

Diffusion coefficient D(θ) and hydraulic conductivity K(θ) are determined by

widely used formulas of van Genuchten [1]-[2]:

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. Discrete analogue of the direct

problem has a form:

148

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−1

i−1/2θni−1 +

1

h2Dn−1

i+1/2θni+1−

−1

τ+

1

h2

(Dn−1

i +Dn−1i−1/2

)θni +

+

θn−1i

τ+

1

h

(Kn−1

i−1/2 −Kn−1i

)= 0,

1 ≤ i ≤ I − 1, 1 ≤ n ≤ N,

ΦnI = θnI − ψn = 0, 1 ≤ n ≤ N, θ0i = ϕi, 0 ≤ i ≤ I.

As a rule, parameters K0, α, m, θmin, θmax are hard-determined through

experiments. Earlier in [3] some of these parameters were determined as a re-

sult of solution of an optimal control problem, in which the objective function is

mean square deviation of simulated soil moisture from some prescribed values.

Numerical solution was determined by gradient method. The gradient of ob-

jective function was calculated applying fast automatic differentiation formulas

[4]. But the process of numerical optimization was difficult. Here, we reformu-

late optimal control problem and introduce new objective function. Obtained

numerical results are discussed and analyzed.

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 hydraulic

conductivity of unsaturated soils,” Soil. Sci. Soc. Am. J., 44, No. 12,

892–898 (1980).

3. E.S. Zasukhina, V.V. Dikusar. “Determining parameters in the model of wa-

ter transfer in soil,” In Abstr. of IV Intern. Conference ”Optimization and

Applications” (OPTIMA-2013), Montenegro, Sept. 22-28, 2013. pp. 169–170.

4. Yu.G. Evtushenko. “Computation of exact gradients in distributed dynamic

systems,” Optimization methods and software, 9, 45–75 (1998).

149

On a variant of dual simplex-like algorithm for linearsemi-definite programming problem

Vitaly Zhadan1


[email protected]

Consider the linear semi-definite programming problem

min C •X, Ai •X = bi, 1 ≤ i ≤ m, X 0, (1)

where C,X and Ai, 1 ≤ i ≤ m, are symmetric matrices of order n, the inequality

X 0 indicates that X must be a semi-definite matrix. The operator • denotes

the Frobenius inner product between two matrices. The dual problem to (1) is

max bTu, Σmi=1u

iAi + V = C, V 0, b = [b1, . . . , bm], u = [u1, . . . , um]. (2)

In [1] the simplex-like algorithm for solving (1) had been proposed. Here the

variant of this method intended for solving the dual problem (2) is considered.

Similar to the primal algorithm the dual one is based on some approach for

solving the system of optimality conditions for (1)

X • V = 0, Ai •X = bi, V = C − Σmi=1u

iAi,

where X 0, V 0, 1 ≤ i ≤ m. These conditions can be rewritten in

vector form, using direct sums of columns of matrices. It is proved that the

dual algorithm generates the sequence of extreme points uk, which belongs

to feasible set of (2) and converges to the solution of (2).

The author was supported by the Russian Academy of Sciences (Program

I.33 P), by the Russian Foundation for Basic Research (project no. 15-01-08259)

and by the Leading Scientific Schools Grant no. 8860.2016.1.

References

1. V.G.Zhadan “On a variant of simplex-like method for linear semi-definite

problem,” Proceedings of the Institute of Mathematics and Mechanics UrB

RAS, 21, no. 3, 117 – 127 (2015) (in Russian).

150

Control of Phase Boundary Evolution in MetalSolidification for New Thermodynamic Parameters of

the Metal

Vladimir Zubov1, Alla Albu1


[email protected]

The problem of controlling the phase boundary evolution in the course of

solidification of metals with different thermodynamic properties is studied. This

problem models the solidification of molten metal in casting. According to

numerous studies of this process, for a product of high quality to be obtained

in a given setup, it is desirable that the shape of the phase boundary be as

close to a plane as possible and that its speed be close to a prescribed one. The

underlying mathematical model of the process is based on a three-dimensional

nonstationary two-phase initialboundary value problem of the Stefan type.

The solidification of the metal can be described as follows. Molten metal

is poured into a mold. The cooling of the mold and the metal inside it occurs

due to the interaction of the object with its surroundings. For this purpose

is used a special setup consisting of a melting furnace, inside which the object

moves, and a cooler which is a large tank filled with liquid aluminum. The mold

with molten metal is being immersed into the coolant. Liquid aluminum has a

relatively low temperature and thus proceeds the crystallization of metal. On

the other hand, the object gains heat from the furnace walls, which prevents

the solidification process from proceeding too fast.

The velocity of the mold relative to the furnace is a parameter that has a

large effect on the evolution of the solidification front in the metal. We use it as a

control function. To find a control function satisfying the imposed technological

requirements, we formulate an optimal control problem. This problem consists

in choosing a regime of metal cooling and solidification in which the solidification

front has a preset shape and moves at a speed close to the preset one.

In this work, the problem of controlling the phase boundary evolution in

151

solidification is considered for a new material to be solidified. The goal of this

research is to analyze the influence exerted on the solution of the optimal control

problem by the thermodynamic parameters of the material under study.

The study was performed for a new improved setup model in which the ceil-

ing is heated up to a rather high temperature. Specifically, this model describes

the situation when several molds are lined up in the furnace near each other

(as occurs in an actual setup). It was assumed that some of the lateral walls

of the mold (on the sides where there are no vertical furnace walls) are made

of a heat-insulated material that loses its thermal insulation property in liquid

aluminum.

For the new material under study, research was carried out concerning the

grids to be used. The time and space grids were chosen so as to achieve the

required accuracy of the numerical results. The time grid was nonuniform. A

finer time grid was used at the beginning of the cooling process, since the largest

variations in the temperature field were observed over this time period. Because

of the new thermodynamic parameters, the time step was smaller (roughly by

25%) than that used with the old parameters.

The control functions are determined by optimal control problems, which

are solved numerically with the help of gradient optimization methods. The

gradient of the cost function is exactly computed by applying the fast automatic

differentiation technique.

Based on the studies performed in this work, we conclude that the thermo-

dynamic properties of materials have a large effect on the solidification process

and that the approach proposed for the control of the phase boundary evolu-

tion in solidification is effective and can be applied to materials with various

thermodynamic properties.

This work was supported by the Basic Research Program of the Presidium

of Russian Academy of Sciences, project no. I.33Π

152

Endogenous Club Formation in a Uni-dimensionalWorld with the Possibility to Stay Alone

Daniil Musatov1,2,3, Alexei Savvateev1,2,4, Anton Trubakov2, Shlomo

Weber1,5

1 New Economic School, Moscow, Russia.2Moscow Institute of Physics and Technology, Russia.

3Kazan Federal University, Russia.4The Dmitry Pozharskiy University, Moscow, Russia.

5 Southern Methodist University, Dallas, USA.

[email protected]; [email protected]; [email protected];

[email protected]

We consider a game-theoretic model of jurisdiction formation where a con-

tinuum of agents vary by a uni-dimensional parameter. The agents may form

communities to enjoy a club good. Its cost depends both on the agent’s char-

acteristic and the club composition. The benefits are substantially small and

agents may choose to stay alone. We show that even in a simple model no

Nash-stable club collection may exist.

THE MODEL

A continuum of agents live on a unit interval [0, 1]. Their population density

f is piecewise continuous and non-zero. If a connected jurisdiction S = [a, b] is

formed then each member gets some benefit V . The cost, however, is distibuted

unevenly. Denote by P (S) the population of S, i.e., P (S) =∫ b

af(x) dx and by

m(S) the median of S, i.e., such point m that∫ m

af(x) dx =

∫ b

mf(x) dx. Then

an agent with characteristic z bears cost c(z, S) = gP (S)

+ ρ|m(S) − z| whilejoining coalition S. This function is defined even if z is not a member of S. The

rationale behind such a cost is the folllowing: the cost g of providing the good is

distributed evenly; the good is situated at m(S) since this point minimizes the

average distance from the agents, and each agent pays ρ for a unit of distance

153

to the good.

D e f i n i t i o n 1. A collection is a finite set of connected jurisdictions

S1 = [a1, b1], . . . , Sn = [an, bn] with pairwise disjoint interiors. W.l.o.g. we may

assume that 0 ≤ a1 < b1 ≤ a2 < b2 ≤ · · · ≤ an < bn ≤ 1.

D e f i n i t i o n 2. A collection is Nash stable if no agent wishes to change

their affiliation. Namely, three inequalities must hold:

(i) Individual rationality: c(z, Sk) ≤ V for any k and z ∈ Sk;

(ii) Stability for members: c(z, Sk) ≤ c(z, Sl) for any k 6= l and z ∈ Sk;

(iii) Stability for non-members: if z does not belong to any Sk then c(z, Sk) ≥V for all k.

ANALYSIS

One may easily obtain the following nice result.

Theorem 1. Any Nash stable collection must satisfy the border indifference

property (BIP): a border agent must be indifferent between the two options.

Namely, if bk = ak+1 then c(bk, Sk) = c(ak+1, Sk+1) and if bk < ak+1 then

c(bk, Sk) = 0 and c(ak+1, Sk+1) = 0. (Similarly, if a1 > 0 then c(a1, S1) = 0

and if bn < 1 then c(bn, Sn) = 0.)

But even border indifference is not guaranteed.

Theorem 2. For certain values of the parameters no non-trivial Nash stable

collection while some jurisdiction is beneficial for all members.

To prove Theorem 2, we construct and carefully analyze a very sophisticated

example with V = 0.1115, ρ = 29and the following density:

f(x) =

45, x ∈[39, 49

]∪[69, 79

];

36, x ∈(49, 59

];

27, otherwise.

We show that S =[39, 79

]is profitable for all members but nevertheless no

nontrivial collection with BIP exist.

The authors wish to acknowledge the support of the Ministry of Education

and Science of the Russian Federation, grant #14.U04.31.0002, administered

154

through the NES Center for Study of Diversity and Social Interactions.

References

1. A. Alesina, E. Spolaore. “On the Number and Size of Nations,” Quarterly

Journal of Economics, 113, 1027–56 (1997).

2. A. Bogomolnaia, M. Le Breton, A. Savvateev, S. Weber. “Stability of jurisdic-

tion structures under the equal share and median rules,” Economic Theory,

34, 525–543 (2008).

3. A. Mas-Colell. “Efficiency and decentralization in the pure theory of public

goods,” Quarterly Journal of Economics, 94, 643–673 (1980).

155

Classical optimal design on annulus

Petar Kunstek1, Marko Vrdoljak2

1 Department of Mathematics, Faculty of Science, University of Zagreb,

Croatia; [email protected] Department of Mathematics, Faculty of Science, University of Zagreb,

Croatia; [email protected]

We optimize a distribution of two isotropic materials that occupy an annulus

in two or three dimensions, heated by a uniform heat source, aiming to maximize

the total energy. In elasticity, the problem models the maximization of the

torsional rigidity of a cylindrical rod with annular cross section made of two

homogeneously distributed isotropic elastic materials [4]. More precisely, we

consider the conductivity problem in an annulus Ω ⊆ Rd:

−div(A∇u) = f

u ∈ H10(Ω) ,

where the conductivity matrix A is of the form A = χαI + (1 − χ)βI, with

a characteristic function χ representing the region occupied by the first phase.

The optimal design problem deals with maximization of the energy functional

I(χ) =∫Ωfu dx, over the set of all measurable characteristic functions χ satis-

fying the condition∫Ωχ dx = qα, which prescribes the amounts of given phases.

Commonly, optimal design problems do not have solutions (such solutions

are called classical), so one considers proper relaxation of the original problem.

Relaxation by the homogenization method consists in introducing generalized

materials, which are mixtures of original materials on the micro-scale.

We shall consider the problem with a constant right-hand side f . The inter-

esting result is that on a simply connected open set Ω, with smooth connected

boundary, the classical solution appears only if Ω is a ball [2], which was origi-

naly showed in [5] but with an additional smoothness assumption on interface

between phases. Moreover, even on a ball, if maximization is replaced by mini-

mization, the optimal design is not classical [6]. For the problem of minimizing

156

energy functional on a ball and arbitrary right-hand side the explicit calcula-

tion of optimal microstructure is presented in [3], while the multiple state case

is treated in [1], and in both situations no classical solution occurs.

If Ω is a ball, in order to maximize the energy the better conductor should be

placed inside a smaller (concentric) ball, whose radius can easily be calculated

from the constraint on given amounts of materials. By analysing the optimality

conditions [5,7], we are able to show that in the case of annulus, the solution is

also unique, classical and radial. Depending on the amounts of given materials,

we find two possible optimal configurations. If the amount of the first phase

is less than some critical value, then the better conductor should be placed in

an outer annulus. Otherwise, the optimal configuration consists of an annulus

with the better conductor, surrounded by two annuli of the worse conductor.

The same holds true in two and three dimensions. The precise solution can

be determined by solving a system of nonlinear equations, which can be done

only numerically.

The second author was supported by the Croatian Science Foundation under

the project 9780 WeConMApp.

References

1. K. Burazin, M. Vrdoljak. “Exact solutions in optimal design problems for

stationary diffusion equation,” under review.

2. J. Casado-Dıaz. “Smoothness properties for the optimal mixture of two

isotropic materials: the compliance and eigenvalue problems,” SIAM J. Con-

trol Optim., 53, 2319–2349 (2015).

3. J. Casado-Dıaz. “Some smoothness results for the optimal design of a two-

composite material which minimizes the energy,” Calc. Var. Partial Differ-

ential Equations, 53, 649–673 (2015).

4. K.A. Lurie, A.V. Cherkaev, A.V. Fedorov. “Regularization of optimal design

problems for bars and plates I-II,” J. Optim. Theory Appl., 37, 499–543

(1982).

157

5. F. Murat, L. Tartar. “Calcul des Variations et Homogeneisation,” in: Les

Methodes de l’Homogenisation Theorie et Applications en Physique, Collect.

Dir. Etudes Rech. Elec. France 57, Eyrolles, Paris, 1985, pp. 319–369.6. L. Tartar. “The Appearance of Oscillations in Optimization Problems,” in:

Non-Classical Continuum Mechanics, London Math. Soc. Lecture Note Ser.

122, Cambridge University Press, 1987, pp. 129–150.7. M. Vrdoljak. “Classical optimal design in two-phase conductivity problems,”

to appear in SIAM Journal on Control and Optimization

158

Large Optimization and Asymptotic Stability

B. Polyak1

1 Institute for Control Science RAS, Moscow, Russia; [email protected]

At first glance, the problems of unconstrained optimization and asymptotic

stability represent quite separate fields of research. However they have much in

common, both in the problem formulation and the techniques exploited. The

unconstrained minimization problem is formulated as follows: Given a function

to be minimized, design an algorithm (in the form of either difference or differ-

ential equation) and prove its convergence to a minimum point. Conversely, the

analysis of asymptotic stability of an equilibrium point of a difference or differ-

ential equation is traditionally performed via use of Lyapunov’s direct method.

Namely, a Lyapunov function is constructed such that it decreases monotoni-

cally on the trajectories of the system. In other words, the given data and the

design goals in both problems change places. We summarize this concept in the

table below:

Problem Optimization Stability

Data Function Equation

Technique Design a method Construct a Lyapunov function

Goal Prove convergence Prove stability

These links between the two fields enable their mutual enrichment both

in terms of techniques and results. First, in the optimization theory, one can

exploit the diversity of Lyapunov functions beyond the standard candidates

(objective function or the distance to a minimum point). We will demonstrate

this possibility for the heavy-ball method in unconstrained minimization. On

the other hand, in the asymptotic stability analysis, it was traditional just to

prove stability, while the rate of convergence was of a lesser interest; moreover,

it cannot be assessed with the commonly used technique based on LaSalle’s in-

variance principle. Needless to say that in optimization, finding such estimates

is the key issue. Also, in optimization problems it is often the case that the

159

minimum point is not unique or may not exist. Similar nonstandard assump-

tions, the non-uniqueness of the equilibrium are of interest in the analysis of

difference and differential equations.

The work is supported by Russian Science Foundation, grant 16-11-10015.

160

Author index

Abgaryan Karine . . . . . . . . . . . . . 10, 12

Abramov Alexander P. . . . . . . . . . . . 14

Abramov Oleg . . . . . . . . . . . . . . . . . . . . 15

Afanas'ev A.P. . . . . . . . . . . . . . . . . 17, 19

Aida-zade Kamil . . . . . . . . . . . . . . . . . 21

Albu Alla . . . . . . . . . . . . . . . . . . . . . . . 151

Anikin Anton . . . . . . . . . . . . . . . . . 50, 68

Antipin Anatoly . . . . . . . . . . . . . . . . . . 23

Atamuratov A.Zh. . . . . . . . . . . . . . . . . 24

Baklanov Artem . . . . . . . . . . . . . . . . . . 26

Beklaryan Armen . . . . . . . . . . . . . . . 144

Belousov Fedor . . . . . . . . . . . . . . . . . . 144

Berezkin Vadim E. . . . . . . . . . . . . . . . 28

Bronnikov Sergey . . . . . . . . . . . . . . . . . 93

Bugerya Alexander . . . . . . . . . . . . . . 146

Bykov Nikita . . . . . . . . . . . . . . . . . . . . . 29

Chentsov Pavel . . . . . . . . . . . . . . . . . . . 26

Chernetsova Daria . . . . . . . . . . . . . . . . 30

Chernov Alexey . . . . . . . . . . . . . . . . . . 32

Chikrii Greta Ts. . . . . . . . . . . . . . . . . . 34

Denisov Dmitry . . . . . . . . . . . . . . . 35, 37

Digo Galina . . . . . . . . . . . . . . . . . . . . . . 38

Digo Natalya . . . . . . . . . . . . . . . . . . . . . 38

Dikusar Vasily . . . . . . . . . . . . . . . . . . . . 40

Druzhinina Olga . . . . . . . . . . . . . 42, 104

Dubinin Roman . . . . . . . . . . . . . . . . . . 87

Dvure hensky Pavel . . . . . . . . . . . . . . 32

Dzyuba S.M. . . . . . . . . . . . . . . . . . 17, 19

Elkin Vladimir . . . . . . . . . . . . . . . . . . . 44

Emelyanova I.I. . . . . . . . . . . . . . . . 17, 19

Eremeev Anton . . . . . . . . . . . . . . . . . . . 46

Erzin Adil . . . . . . . . . . . . . . . . . . . . . . . . 47

Evtushenko Yuri G. . . . . . . . . . . . . . . 48

Eyniev El hin . . . . . . . . . . . . . . . . . . . . 66

Finkelstein Evgeniya . . . . . . 50, 51, 68

Førsund Finn . . . . . . . . . . . . . . . . . . . . 92

Galashov Alexander . . . . . . . . . . . . . . 53

Ganin Bogdan V. . . . . . . . . . . . . . . . . . 55

Garanzha V.A. . . . . . . . . . . . . . . . . . . . 57

Gasnikov Alexander . . . . . . . . . . . . . . 32

Gerasimov Aleksey . . . . . . . . . . . . . . . 93

Giannessi Fran o . . . . . . . . . . . . . . . . . 59

Gimadi Edward . . . . . . . . . . . . . . . . . . 60

Golikov Alexander I. . . . . . . . . . . . . . 48

Golshteyn Evgeny . . . . . . . . . . . . . . . . 62

Gon harov Evgenii . . . . . . . . . . . . . . . 64

Gon harov Vasily . . . . . . . . . . . . . . . . . 66

Gordeeva Valeriya . . . . . . . . . . . . . . . 136

Gornov Alexander . . . . . . . . 26, 51, 68

Gruzdeva Tatiana . . . . . . . . . . . . . . . . 69

Guliyev Samir . . . . . . . . . . . . . . . . . . . . 21

Istomin Alexey . . . . . . . . . . . . . . . . . . . 60

Ja imovi Miloji a . . . . . . . . . . . . . . . 72

Ja imovi Vladimir . . . . . . . . . . . . . . . 71

Kaporin Igor . . . . . . . . . . . . . . . . . . . . . 74

161

Karamzin Dmitry . . . . . . . . . . . . . . . . 76

Karpenko Anatoly . . . . . . . . . . . . . . . 124

Kel'manov Alexander . 46, 53, 78, 80,

8284

Kha haturov Ruben V. . . . . . . . . . . . 85

Kha hay Mi hael . . . . . . . . . . . . . . 8789

Khamidullin Sergey . . . . . . . 80, 82, 84

Khandarov Fedor . . . . . . . . . . . . . . . . 146

Khandeev Vladimir . . . . . . . . . . . 80, 82

Kharlamov Maxim . . . . . . . . . . . . . . . 93

Khoroshilova Elena . . . . . . . . . . . . . . . 90

Khusnullin Nail . . . . . . . . . . . . . . . . . . 93

Konatar Nikola . . . . . . . . . . . . . . . . . . . 71

Krivonozhko Vladimir . . . . . . . . . . . . 92

Kudryavtseva L.N. . . . . . . . . . . . . . . . 57

Kunstek Petar . . . . . . . . . . . . . . . . . . 156

Latiy Vladislav . . . . . . . . . . . . . . . . . . . 37

Lazarev Alexander . . . . . . . . . . . . 93, 95

Lebedev Konstantin . . . . . . . . . . 97, 98

Lebedev Valery . . . . . . . . . . . . . . . . . . . 97

Les hov Alexey . . . . . . . . . . . . . . . . . . . 99

Lotov Alexander V. . . . . . . . . . . . . . 101

Lotova Elizaveta A. . . . . . . . . . . . . . . 28

Ly hev Andrey . . . . . . . . . . . . . . . . . . . 92

Malkov Ustav . . . . . . . . . . . . . . . . . . . . 62

Malkova Vlasta . . . . . . . . . . . . . . . . . 102

Malyshev D.I. . . . . . . . . . . . . . . . . . . . 121

Mamaev Y.A. . . . . . . . . . . . . . . . . . . . 121

Masina Olga . . . . . . . . . . . . . . . . . . . . 104

Mijajlovi Nevena . . . . . . . . . . . . . . . . 72

Mikhailov I.E. . . . . . . . . . . . . . . . . . . . . 24

Mikhailova Ludmila . . . . . . . . . . . 80, 82

Min henko Leonid . . . . . . . . . . . . . . . . 99

Misur Marin . . . . . . . . . . . . . . . . . . . . . 106

Mitrovi Darko . . . . . . . . . . . . . . . . . . 106

Motkova Anna . . . . . . . . . . . . . . . . . . . 83

Muravey Leonid . . . . . . . . . . . . . . 24, 66

Musatov Daniil . . . . . . . . . . . . . . . . . . 153

Musatova Elena . . . . . . . . . . . . . . 93, 95

Neznakhina Katherine . . . . . . . . . . . . 88

Noor Muhammad Aslam . . . . . . . . . 72

Novak Andrej . . . . . . . . . . . . . . . . . . . 106

Obrosova Nataliia . . . . . . . . . . . . . . . 107

Olenev Ni holas . . . . . . . . . . . . . . . . . 109

Parkhomenko Valeriy . . . . . . . . . . . . 111

Pereira Fernando . . . . . . . . . . . . . . . . . 76

Pesterev Alexander . . . . . . . . . . . . . . 113

Petrov Aleksey . . . . . . . . . . . . . . . . . . . 93

Petrov Lev . . . . . . . . . . . . . . . . . . . . . . 115

Petrova Natalia . . . . . . . . . . . . . . . . . . 42

Petrovykh Aleksandr . . . . . . . . . . . . 116

Poberiy Maria . . . . . . . . . . . . . . . . . . . . 89

Polyak B. . . . . . . . . . . . . . . . . . . . . . . . 159

Ponomarev Konstantin . . . . . . . . . . . 93

Popov Leonid D. . . . . . . . . . . . . . . . . 117

Prusinska Agnieszka . . . . . . . . . . . . . 119

Putilina E.V. . . . . . . . . . . . . . . . . . 17, 19

Pyatkin Artem . . . . . . . . . . . . . . . . . . . 46

Rentsen Enkhbat . . . . . . . . . . . . . . . . . 50

162

Roman henko Semyon . . . . . . . . . . . . 84

Ryabikov Andrey I. . . . . . . . . . . . . . 101

Rybak L.A. . . . . . . . . . . . . . . . . . . . . . 121

Sakharov Maxim . . . . . . . . . . . . . . . . 124

Savelyev V.P. . . . . . . . . . . . . . . . . . . . 127

Savvateev Alexei . . . . . . . . . . . . . . . . 153

Shamin A.A. . . . . . . . . . . . . . . . . . . . . 127

Shananin Alexander . . . . . . . . . . . . . 107

Shmyrev V.I. . . . . . . . . . . . . . . . . . . . . 129

Skarin Vladimir D. . . . . . . . . . . . . . . 117

Skiba Alexander . . . . . . . . . . . . . . . . . 130

Sokolov Nikolay . . . . . . . . . . . . . . . . . . 62

Strekalovsky Alexander . . . . . . 69, 132

Svinin Mikhail . . . . . . . . . . . . . . . . . . . 51

Tarasov Ilia . . . . . . . . . . . . . . . . . . . . . . 95

Tikhomirov Nikolay . . . . . . . . . . . . . 134

Tikhomirova Tatiana . . . . . . . . . . . . 136

Tret'yakov Alexey . . . . . . . . . . . . . . . 119

Trubakov Anton . . . . . . . . . . . . . . . . . 153

Tsidulko Oxana Yu. . . . . . . . . . 60, 138

Turkin Andrei . . . . . . . . . . . . . . . . . . . 140

Tyatyushkin Alexander . . . . . . . . . . . 68

Tyupikova Tatyana . . . . . . . . . . . . . . . 98

Uvarov Sergey . . . . . . . . . . . . . . . . . . . . 10

Vasin Alexander . . . . . . . . . . . . . . . . . 142

Vlasova Nataly . . . . . . . . . . . . . . . . . . . 29

Volodina Olga . . . . . . . . . . . . . . . . . . . . 12

Vrdoljak Marko . . . . . . . . . . . . . . . . . 156

Weber Shlomo . . . . . . . . . . . . . . . . . . 153

Wojtowi z Marek . . . . . . . . . . . . . . . . . 40

Yadrentsev Denis . . . . . . . . . . . . . . . . . 93

Zarodnyuk Tatiana . . 26, 68, 144, 146

Zasukhin Sergey . . . . . . . . . . . . . . . . . 148

Zasukhina Elena . . . . . . . . . . . . . 40, 148

Zhadan Vitaly . . . . . . . . . . . . . . . . . . 150

Zinder Yakov . . . . . . . . . . . . . . . . . . . . . 95

Zubov Vladimir . . . . . . . . . . . . . . . . . 151

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