Network design with equilibrium-driven customer demands ... · PDF fileNetwork design with equilibrium-driven customer demands Mahdi Hamzeei and James Luedtke University of Wisconsin-Madison

Network design with equilibrium-driven customer demands

Mahdi Hamzeei and James LuedtkeUniversity of Wisconsin-Madison

N1

M2

M1

N4

M3

N3

Given:

N: set of customers

M: set of facilities

Demand of customer i 2 N

Capacity of facility j 2 M

Cost of travel from i to j

Cost of opening facility j

The system operator wishes to open a subset of facilitiesto open to balance customer service with cost of openingthe facilities.

Given a subset of open facilities, customers choose towhich facilities to send their demand.

Each facility faces congestion which depends on theamount of demand sent to the facility.

Customers choose facilities that minimize travel andcongestion cost of the facility.

Key challengeThe allocation of demands of customers to facilities is thesolution of an equilibrium constraint.

Hamzeei and Luedtke University of Wisconsin-Madison 1/2

Proposed approachFormulated as a mixed-integer bilevel programProposed a solution method:

1 Derived a Lagrangian relaxation2 Lagrangian dual problem can be decomposed into

subproblems, one for each facility3 Subproblems can be solved efficiently.4 Solution of the relaxation is used to obtain a feasible solution

to the original problem

Gap between lower and upper bound indicates the strength ofthe solution

Hamzeei and Luedtke University of Wisconsin-Madison 2/2

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Computational investigation of generalized intersection cuts

Aleksandr M. Kazachkov!Tepper School of Business, Carnegie Mellon University

Based on joint work with Egon Balas, François Margot, and Selvaprabu Nadarajah

How to choose hyperplanes?

Best strategy for generating cuts?

DECOMPOSITION ALGORITHMS FOR TWO-STAGECHANCE-CONSTRAINED PROGRAMS

Xiao Liu, Simge Kucukyavuz and James Luedtke

When we are making decisions about uncertain future:

Can we combine these two models?

I Two-stage CCMP : consider costs of recourse actions for feasiblescenarios.

I Two-stage CCMP with Recovery (CCMPR) : model the needs to“recover” from a infeasible scenario.

1 / 2

Solution algorithm

What has been done:

However, a simple combination of these two approaches does not yield goodresults for 2-stage CCMPR.

What we do:

I Derive a class of strong valid optimality cuts.I Propose a Benders-type decomposition algorithm.

2 / 2

Open-source algebraic modeling in Julia

Miles LubinMIT Operations Research Center

MIP 2014

1 / 3

State of the art: AMPL, GAMS, Pyomo, CPLEX Concert, CVX,YALMIP, PuLP, GurobiPy, Gurobi C++, CMPL, AIMMS, ZIMPL,...

In MIP research we want:

I Ease of use (Python/MATLAB over C/C++)

I Access to low-level solver (callbacks)

I Performance

I (Solver independence)

Can we have it all?

2 / 3

A new modeling language for optimization

I Embedded in Julia, a high-performance high-level language fortechnical computing

I Familiar syntax for users of Python/MATLAB/AMPL

I As fast as hand-written sparse matrix generatorsI Solver-independent B&B callbacks for implementing cutting

planes (and more)I Gurobi, CPLEX, GLPK

I Derivatives and expression trees for MINLP (without AMPL)

I Open source, easy to extend

3 / 3

A machine learning based approximation of

strong branching

in the 2014 Mixed Integer Programming Workshop (MIP 2014)

Alejandro Marcos Alvarez Quentin Louveaux Louis Wehenkel

University of Liege

July 21, 2014

1

Motivation

Strong branching generally

minimizes the number of nodes of the tree

is intractable in practice

Idea ! approximate strong branching with a more tractable function

This is not new !

Innovation of our approach: we use machine learning techniques

2

What you will see on our poster

1

discover what machine learning is

2

how machine learning can be used to construct a branching

heuristic

3

discover a set of features describing fractional variables (main

part of the work)

3

The$Power$of$a$Negative$Eigenvalue:$Aggregation$Cuts$for$Nonlinear$Integer$

Programming$!

Sina$Modaresi$MIT/University.of.Pittsburgh.

!

Joint.Work.with.Juan.Pablo.Vielma.(Massachusetts.Institute.of.Technology).

!

Mixed.Integer.Programming.Workshop,.Columbus,.July.2014.

• “Extended”.Aggregation.Technique.(Yildiran,.2009).

• Characterizing.convex.hull.of.a.set.defined.by.two.quadratic.inequalities.

!

•No.specific.requirement.on.the.two.quadratics.

• Convex.hull.characterization.is.independent.of.the.geometry.of.the.set

AggregaTon.Cuts.for.MINLP

Monday,.July.21,.2014.............The.Power.of.a.NegaTve.Eigenvalue:.AggregaTon.Cuts.for.Nonlinear.Integer.Programming. ..............................

S :=�x 2 Rn : xT

Qix+ 2bTi x+ �i < 0, i = 1, 2

•



The Power of a Negative Eigenvalue: Aggregation Cuts for

Nonlinear Integer Programming

Sina Modaresi

⇤Juan Pablo Vielma

†

July 20, 2014

S := {x 2 Rn: qi < 0, i = 1, 2} ,

where

qi = x

TQix+ 2b

Ti x+ �i i = 1, 2.

Example. Let

S :=

�

(x, t) 2 R2: x

2 � t

2+ 2x+ 2 0, �x

2+ t

2+ 2x� 2 0

.

q =

x

1

�T

P

x

1

�

, (1)

where x 2 Rn, P =

Q b

b

T�

�

2 Sn+1, Q 2 Sn

, b 2 Rn, and � 2 R. The above quadratic

function can also be expressed as

q = x

TQx+ 2b

Tx+ �.

They study the set S defined by two quadratic inequalities as

S := {x 2 Rn: qi < 0, i = 1, 2} .

Denote the quadratic inequality induced by the convex combination of the above quadratic

inequalities as

q� := (1� �)q1 + �q2,

where � 2 [0, 1]. Similarly, define

P� := (1� �)P1 + �P2 and Q� := (1� �)Q1 + �Q2.

Also define the associated quadratic set as

S� := {x 2 Rn: q� < 0} .

⇤Industrial Engineering Department, University of Pittburgh, Pittsburgh PA 15261, [email protected]

†Sloan School of Management, Massachusetts Institute of Technology, Cambridge MA 02139, [email protected]

1

Sina Modaresi

•Characterizing.convex.hull.of.a.set.defined.by.a.conic.and.a.quadratic.inequality.

!

!

.......using.the.same.aggregation.technique.

!



The Power of a Negative Eigenvalue: Aggregation Cuts for

Nonlinear Integer Programming

Sina Modaresi

⇤Juan Pablo Vielma

†

July 20, 2014

S := {x 2 Rn: kAx� ck2 < a

Tx� d

x

TQx+ 2b

Tx+ � < 0},

S := {x 2 Rn: qi < 0, i = 1, 2} ,

where

qi = x

TQix+ 2b

Ti x+ �i i = 1, 2.

Example. Let

S :=

�

(x, t) 2 R2: x

2 � t

2+ 2x+ 2 0, �x

2+ t

2+ 2x� 2 0

.

q =

x

1

�T

P

x

1

�

, (1)

where x 2 Rn, P =

Q b

b

T�

�

2 Sn+1, Q 2 Sn

, b 2 Rn, and � 2 R. The above quadratic

function can also be expressed as

q = x

TQx+ 2b

Tx+ �.

They study the set S defined by two quadratic inequalities as

S := {x 2 Rn: qi < 0, i = 1, 2} .

Denote the quadratic inequality induced by the convex combination of the above quadratic

inequalities as

q� := (1� �)q1 + �q2,

where � 2 [0, 1]. Similarly, define

P� := (1� �)P1 + �P2 and Q� := (1� �)Q1 + �Q2.

⇤Industrial Engineering Department, University of Pittburgh, Pittsburgh PA 15261, [email protected]

†Sloan School of Management, Massachusetts Institute of Technology, Cambridge MA 02139, [email protected]

1

On Blocking and Anti-Blocking Polyhedra in Infinite Dimensions

Luis Rademacher Alejandro Toriello Juan Pablo Vielma

Motivating Question • For infinite horizon planning:

Infinite objective function can be avoided by using either – a discount 𝛼 ∈ 0,1

𝛼 𝑐 𝑥

– a cumulative average

lim sup

→

1𝑛 𝑐 𝑥

• Drawbacks: distant future or near future have low influence,

respectively.

Our results

• Use complementary slackness as a notion of optimality with (possibly) infinite objective value: For packing-covering problems (blocking anti-blocking polyhedra): – existence of optimal primal-dual pairs – equivalent to time-localized optimality – integrality of extreme points for natural problems.

MILP model for a real-world hydro-powerunit-commitment problem

Pascale Bendotti1, Claudia D’Ambrosio2, Grace Doukopoulos1,Arnaud Lenoir1, Leo Liberti2 Youcef Sahraoui1,2

1OSIRIS, EDF R&D

2LIX, Ecole Polytechnique

MIP workshop - Ohio State UniversityJuly 21st, 2014

(EDF R&D, Ecole Polytechnique) Short-term hydro scheduling 05/06/2014 1 / 3

Challenge

Solve a large-scale MILP with real-world data.J

Goalsconvergence to optimalitylimited resolution time-framemore refined model to fit real description of operationsN

Obstaclesinfeasibilitiesnumerical difficultieslong resolution times and intractabilities

=) idenfify and cope with obstacles


Approach, 1st results and future work

1 Approach:Simplify model by using notably less binary variables to get firstunderstanding

2 Results for infeasibilities (with Cplex conflict refiner):discard (or modify) datarelax or penalize constraints (or consider as 2nd objective)

3 Future work:dealing with numerical instabilitiesunderstanding poor resolution performancesolving the real problem: bi-objective, decomposition


• Master Knapsack Problem K(n) is the integer solutions to

t1+ 2 t2 + … + n tn = n, “the knapsack equation”

!• Knapsack facets ξt ≤ 1 (assume ξ1 = 0.)

!• A knapsack facet is a 1/k-facet if coefficients

ξi ϵ { 0/k, 1/k, …, k/k } U { ½ } for all i = 1, …, n.

A Few Strong Knapsack Facets - Shim, Cao, Chopra, Steffy

1

• Subproblem = general integer knapsack problem

!• Every integer knapsack problem is a projection of K(n) ⇔ 5 t

5 + 6 t

6 + 4 t

4 ≤ 10

⇔ 5 t5 + 6 t

6 + 4 t

4 + t

1 = 10

⇔ 1 t1+ 2 t

2 + … + 10 t

10 = 10 and t

2 = t

3 = t

7 = t

8 = t

9 = t

10 = 0

!• A knapsack facet ξ.t ≤ 1 is projected into a valid inequality

ξ{4,5,6}t{4,5,6} ≤ 1

!• Assume l << n.


2

• Separate > 75% of facets in O(l4 = lk/2) !

– Independent of n

!

–Small k are good enough


3

Facets of the bound-relaxed

two-integer knapsack

Ricardo Fukasawa, Laurent Poirrier,

´

Alinson Xavier

Department of Combinatorics and Optimization

University of Waterloo, Canada

July 2014

1 / 3

Introduction and Two-row model

⇧ We study the facial structure of the polyhedron

P = conv

((x , s) 2 Z⇥ Rn

+ : x = f +

nX

i=1

r

i

s

i

, s1 2 Z),

where n 2 Z+, f 2 Q \ Z and r 2 Qn

.

⇧ Two-integer knapsack, non-negativity of an integer variable is relaxed.

⇧ Single-row corner relaxation, integrality of a non-basic variable is kept.

⇧ Can provide cutting planes for general mixed-integer problems.

⇧ Let S = Z⇥ Z+. We can rewrite

P =

((x , s) 2 S ⇥ Rn

+ :

�x1

x2

�=

�f

0

�+

�r1

1

�s1 +

nX

i=2

�r

i

0

�s

i

).

⇧ Correspondence between S-free sets and valid inequalities for P .

2 / 3



P = conv

((x , s) 2 Z⇥ Rn

+ : x = f +

nX

i=1

r

i

s

i

, s1 2 Z),


.





P =

((x , s) 2 S ⇥ Rn

+ :

�x1

x2

�=

�f

0

�+

�r1

1

�s1 +

nX

i=2

�r

i

0

�s

i

).


2 / 3



P = conv

((x , s) 2 Z⇥ Rn

+ : x = f +

nX

i=1

r

i

s

i

, s1 2 Z),


.





P =

((x , s) 2 S ⇥ Rn

+ :

�x1

x2

�=

�f

0

�+

�r1

1

�s1 +

nX

i=2

�r

i

0

�s

i

).


2 / 3



P = conv

((x , s) 2 Z⇥ Rn

+ : x = f +

nX

i=1

r

i

s

i

, s1 2 Z),


.





P =

((x , s) 2 S ⇥ Rn

+ :

�x1

x2

�=

�f

0

�+

�r1

1

�s1 +

nX

i=2

�r

i

0

�s

i

).


2 / 3

Main Results

⇧ All facet-defining inequalities come from:

⇧ Maximal splits unbounded along the line

�f

0

�+ �

�r1

1

�

⇧ Maximal S-free wedges with apex on this same line.

⇧ At most one wedge for each facet of

A = conv{x 2 Z⇥ Z+ : x1 � r1x2 f }, orB = conv{x 2 Z⇥ Z+ : x1 � r1x2 � f }.

⇧ Algorithms to:

⇧ Enumerate the facets of A,B , therefore all facet-defining wedges.

⇧ Calculate trivial lifting coe�cients for additional integer variables.

⇧ Wedge cuts generalize MIR and 2-step MIR cuts.

⇧ Upper bound on the split rank.

⇧ Computational experiments.

3 / 3

Main Results



�f

0

�+ �

�r1

1

�




⇧ Algorithms to:






3 / 3

Main Results



�f

0

�+ �

�r1

1

�




⇧ Algorithms to:






3 / 3

Main Results



�f

0

�+ �

�r1

1

�




⇧ Algorithms to:






3 / 3

Main Results



�f

0

�+ �

�r1

1

�




⇧ Algorithms to:






3 / 3

Generating Multi-row Simplex Cuts on HigherDimensional Spaces

Emre Yamangil 1

1Rutgers University Center for Operations Research

July 21st, 2014

Joint work with Endre Boros

Emre Yamangil (RUTCOR) Multi-row simplex cuts July 21st, 2014 1 / 6

Outline

1 Corner polyhedron

2 Maximal lattice-free convex sets

3 Numerical experiment


Preliminaries

Corner Polyhedron

Let the corner polyhedron be represented as follows:

C = conv

(✓x

s

◆2 Zm ⇥ Rk : x = f +

kX

i=1

ri si , s � 0

)(1)

We want to generate a deepest cut in the sense that it minimizes the sum

of its coe�cients and separates a fractional vector f and C .

Intersection Cut

Let S ⇢ Rm be closed, convex and s.t. f 2 int(S) and8x 2 Zm ) x /2 int(S). Then,

Pkj=1�S(rj)sj � 1 is valid for (1) that cuts

o↵ basic solution f , where �S(rj) := inf{t > 0 : f + rj/t 2 S}, 8j .


Maximal lattice-free convex sets

Theorem (Bell and Scarf (1977))

Let {x 2 Zm : Ax b} = ; where A 2 R`⇥m, then there are 2m or less

inequalities among Ax b, say A

0x b

0, such that

{x 2 Zm : A0x b

0} = ;.

Theorem (Lovasz (1989))

If S is maximal-lattice free convex set on Rms.t. f 2 int(S), then

S = {x 2 Rm : A(x � f ) 1} for some A 2 R`⇥m, where ` 2m can be

assumed.


How to generate lattice-free convex sets

Master Problem

min

Pki=1 �i

s.t. Ari � �i1 0 i = 1, . . . , kmax

i=1,...,l(A(x � f ))i � 1 8x 2 X ⇢ Zm

A 2 R`⇥m

�i � 0 i = 1, . . . , k(2)

Note: Problem (2) is a Fixed-DimensionalSemi-Infinite Mixed-Integer DisjunctiveProgram for fixed number of simplex rowswhere ` 2m.

Separation Problem

max ✏s.t.

¯

A

ix +e✏ 1 +

¯

A

if

x 2 Zm

✏ � 0

(3)

Note: Here A

i denote theoptimal solution ofProblem (2) at iteration i

for set X ⇢ Zm.

Let � be an optimal solution to Problem (2) and optimum objective valueof the corresponding separation problem is 0. Then,

Pkj=1 �jsj � 1 is a

valid inequality for (1).


Diamonds in the rough

Figure : aflow30a

Figure : protfold

Figure : sp97ar

Figure : mzzv11


Two-Term Disjunctions on the Second-OrderCone

Sercan Yıldı[email protected]

joint work withFatma Kılınç-Karzan Gérard Cornuéjols

July 21, 2014

Mixed-integer conic set: S := {x 2K : Ax = b, x

j

2 Z 8j 2 J}.Two-term disjunctive relaxation: C

1

[C

2

where C

i

:= {x 2K : Ax = b, c

>i

x � c

i,0}.

Strong disjunctive cuts for S through a study of conv(C1

[C

2

)

S C

1

[C

2

conv(C1

[C

2

)

Main contribution: Explicit and simple description of conv(C1

[C

2

)when K is the second-order cone.

Sercan Yıldız Two-Term Disjunctions on the Second-Order Cone 1

Mixed-integer conic set: S := {x 2K : Ax = b, x

j

2 Z 8j 2 J}.Two-term disjunctive relaxation: C

1

[C

2

where C

i

:= {x 2K : Ax = b, c

>i

x � c

i,0}.

Strong disjunctive cuts for S through a study of conv(C1

[C

2

)

S C

1

[C

2

conv(C1

[C

2

)

Main contribution: Explicit and simple description of conv(C1

[C

2

)when K is the second-order cone.


PART 1: C

1

[C

2

where C

i

= {x 2K : c

>i

x � c

i,0}.

We derive a family of convex valid inequalities that characterizeconv(C

1

[C

2

):

2c

2,0 � (bc

1

+ c

2

)>x r((bc

1

� c

2

)>x)2 +N (b )⇣

x

2

n

�kxk2

2

⌘.

We show that these inequalities admit an SOC expression under asuitable “disjointness" assumption:

N (b )x+2(c>2

x� c

2,0)

✓b c

1

� c

2

�bc

1,n + c

2,n

◆2K.

We identify cases where a single one of these inequalities issufficient to describe conv(C

1

[C

2

).

PART 2: C

1

[C

2

where C

i

= {x 2K : Ax = b, c

>i

x � c

i,0}.

We identify a set of conditions under which a single inequality ofthe type above is sufficient to describe conv(C

1

[C

2

).

In particular, we can characterize the convex hull of

all two-term disjunctions on ellipsoids and paraboloids,all split disjunctions on ellipsoids, paraboloids, andhyperboloids.


PART 1: C

1

[C

2

where C

i

= {x 2K : c

>i

x � c

i,0}.


1

[C

2

):

2c

2,0 � (bc

1

+ c

2

)>x r((bc

1

� c

2

)>x)2 +N (b )⇣

x

2

n

�kxk2

2

⌘.


N (b )x+2(c>2

x� c

2,0)

✓b c

1

� c

2

�bc

1,n + c

2,n

◆2K.


1

[C

2

).

PART 2: C

1

[C

2

where C

i

= {x 2K : Ax = b, c

>i

x � c

i,0}.


1

[C

2

).




PART 1: C

1

[C

2

where C

i

= {x 2K : c

>i

x � c

i,0}.


1

[C

2

):

2c

2,0 � (bc

1

+ c

2

)>x r((bc

1

� c

2

)>x)2 +N (b )⇣

x

2

n

�kxk2

2

⌘.


N (b )x+2(c>2

x� c

2,0)

✓b c

1

� c

2

�bc

1,n + c

2,n

◆2K.


1

[C

2

).

PART 2: C

1

[C

2

where C

i

= {x 2K : Ax = b, c

>i

x � c

i,0}.


1

[C

2

).




Two-Term Disjunctions on the Second-OrderCone

Sercan Yıldı[email protected]

joint work withFatma Kılınç-Karzan Gérard Cornuéjols

July 21, 2014

Documents

Network design with equilibrium-driven customer demands ... · PDF fileNetwork design with equilibrium-driven customer demands Mahdi Hamzeei and James Luedtke University of Wisconsin-Madison