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A class of valid inequalities for multilinear 0-1 optimization problems

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A class of valid inequalities for multilinear 01

optimization problems

Elisabeth Rodríguez-Heck and Yves Crama

QuantOM, HEC Management School, University of Liège Partially supported by Belspo - IAP Project COMEX

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Multilinear 0-1 optimization

Multilinear 0-1 optimization minX S∈S aS Y i ∈S xi+ l (x ) s. t. xi ∈ {0, 1} i =1, . . . , n

• S: subsets of {1, . . . , n} with aS 6=0 and |S| ≥ 2,

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Standard Linearization (SL)

Standard Linearization minX S∈S aSyS+ l (x ) s. t. yS ≤ xi ∀i ∈ S, ∀S ∈ S yS ≥ X i ∈S xi− (|S| −1) ∀S ∈ S (yS =Qi ∈Sxi)

for variables xi, yS ∈ {0, 1}, the convex hull of feasible solutions is PSL,

for continous variables xi, yS ∈ [0, 1], the set of feasible solutions is PSL. SL drawback: The continous relaxation given by the SL is very weak!

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The 2-link inequalities

Denition

For S, T ∈ S and yS, yT such that yS =

Q

i ∈Sxi, yT =

Q

i ∈Txi,

• the 2-link associated with (S,T ) is the linear inequality

yS≤yT−Pi∈T\Sxi+ |T\S|

• PSL2links is the polytope dened by the SL inequalities and the 2-links. Interpretation

S T

yS =1 ⇒ ∀i ∈ S, xi =1

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Theoretical contributions

Theorem 1: A complete description for the case of two monomials

For the case of two nonlinear monomials, P∗

SL= PSL2links, i.e., the

standard linearization and the 2-linksprovide acomplete

description of P∗

SL.

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Proof idea (Theorem 1):

Consider the extended formulation (with variables in [0, 1])

yS∩T ≤ xi, ∀i ∈ S ∩ T , (1) yS∩T ≥ X i ∈S∩T xi− (|S ∩ T | −1), (2) yS ≤ yS∩T, (3) yS ≤ xi, ∀i ∈ S\T , (4) yS ≥ X i ∈S\T xi+ yS∩T− |S\T |, (5) yT≤ yS∩T, (6) yT≤ xi, ∀i ∈ T \S, (7) yT≥ X i ∈T \S xi+ yS∩T− |T \S|, (8)

Notice that the two polytopes P0 and P1 obtained by xing variable yS∩T

to 0 and 1, resp., are integral.

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Theoretical contributions

Theorem 2: Facet-dening inequalities for the case of two monomials

For the case of two nonlinear monomials dened by S, T with |S ∩ T | ≥2, the 2-links are facet-deningfor P∗

SL.

Proof idea (Theorem 2): Since P∗

SL is full-dimensional (dim n + 2), nd n + 1 anely independent points in the faces dened by the 2-links.

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Computational experiments: are the 2-links helpful for the

general case?

Objectives

• compare thebounds obtained when optimizing over PSL and PSL2links,

• compare thecomputational performance of exact resolution

methods.

Software used: CPLEX 12.06. Inequalities used

• SL: standard linearization (model),

• cplex: CPLEX automatic cuts,

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Random instances: bound improvement

LP relax. gap (%) xed degree

rf-a rf-b rf-c rf-d rf-e rf-f rf-g rf-h 0 2 4 6 8 10 SL SL&2L Fixed degree: inst. d n m rf-a 3 400 800 rf-b 3 400 900 rf-c 3 600 1100 rf-d 3 600 1200 rf-e 4 400 550 rf-f 4 400 600 rf-g 4 600 750 rf-h 4 600 800 8 / 17

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Random instances: computation times results

Run times (in sec.) xed degree

0 100 200 300 400 500 600 700 rf-a rf-b rf-c rf-d rf-e rf-f rf-g rf-h SL SL&2L SL&cplex

SL&cplex&2L Fixed degree:

inst. d n m rf-a 3 400 800 rf-b 3 400 900 rf-c 3 600 1100 rf-d 3 600 1200 rf-e 4 400 550 rf-f 4 400 600 rf-g 4 600 750 rf-h 4 600 800

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Instances inspired from image restoration: denition

0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 Image restoration 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 Base images:

• top left rect. (tl),

centre rect. (cr), • cross (cx). Perturbations: • none (n), • low (l), • high (h).

Up to n = 225 variables and m = 1598 terms

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Image restoration instances: bounds results

LP relax. gap (%) 10x15 images

tl-n tl-l tl-h cr-n cr-l cr-h cx-n cx-l cx-h 0 500 1,000 1,500 SL SL&2L

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Image restoration instances: bounds results

LP relax. gap (%) 15x15 images

tl-n tl-l tl-h cr-n cr-l cr-h cx-n cx-l cx-h 0 200 400 600 800 1,000 1,200 1,400 1,600 SL SL&2L 12 / 17

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Image restoration instances: computation times results

Run times (in sec.) 10x15 images

0 10 20 30 40 50 tl-n tl-l tl-h cr-n cr-l cr-h cx-n cx-l cx-h SL&cplex SL&cplex&2L

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Image restoration instances: computation times results

Run times (in sec.) 15x15 images

0 20 40 60 80 100 120 tl-n tl-l tl-h cr-n cr-l cr-h cx-n cx-l cx-h SL&cplex SL&cplex&2L 14 / 17

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When is SL a complete description?

Summary

• SL + 2-links = a complete description (two nonlinear monomials). • 2-links help computationally for the general case.

Question:

Can we characterize when the SL alone is a complete description of the convex hull P∗

SL?

Joint work with C. Buchheim.

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SL complete description

Multilinear 0-1 optimization minX S∈S aS Y i ∈S xi+ l (x ) s. t. xi ∈ {0, 1} i =1, . . . , n

Standard linearization constraints

yS ≤ xi ∀i ∈ S, ∀S ∈ S yS ≥

X

i ∈S

xi− (|S| −1) ∀S ∈ S

Subsets S dene a hypergraph H.

We write PSL = P

(H) SL .

Matrix of constraints MH.

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SL complete description

Theorem 3

Given a hypergraph H, the following statements are equivalent:

(a) PSL(H) is an integer polytope.

(b) MH is balanced.

(c) H is Berge-acyclic.

Derived from a more general result taking into account the sign pattern of the monomials.

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Some references I

E. Balas Disjunctive programming: properties of the convex hull of feasible points. GSIA Management Science Research Report MSRR 348, Carnegie Mellon University (1974). Published as invited paper in Discrete Applied Mathematics 89 (1998) 344 C. Berge and E. Minieka. Graphs and hypergraphs, volume 7. North-Holland publishing company, Amsterdam, 1973.

Y. Crama. Concave extensions for nonlinear 01 maximization problems. Mathematical Programming 61(1), 5360 (1993)

Y. Crama and E. Rodríguez-Heck. A class of new valid inequalities for multilinear 0-1 optimization problems. Submitted for publication, 2016.

C. Buchheim, Y. Crama and E. Rodríguez-Heck. Berge-acyclic multilinear 01 optimization problems. Manuscript, 2016.

C. De Simone. The cut polytope and the boolean quadric polytope. Discrete Mathematics, 79(1):7175, 1990.

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Some references II

R. Fortet. L'algèbre de Boole et ses applications en recherche opérationnelle. Cahiers du Centre d'Études de recherche opérationnelle, 4:536, 1959.

F. Glover and E. Woolsey. Further reduction of zero-one polynomial programming problems to zero-one linear programming problems. Operations Research, 21(1):156161, 1973.

P. Hansen and B. Simeone. Unimodular functions. Discrete Applied Mathematics, 14(3):269281, 1986.

M. Padberg. The boolean quadric polytope: some characteristics, facets and relatives. Mathematical Programming, 45(13):139172, 1989.

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SL complete description: signed case

Theorem 4

Given a hypergraph H = (V , E) and a sign pattern s ∈ {−1, 1}E, the

following statements are equivalent:

(a) For all f ∈ P(H) with sign pattern s, every vertex of PH

maximizing Lf is integer.

(b) MH(s) is balanced.

(c) H(s) has no negative special cycle.

(d) PH(s)is an integer polytope. PH(s) is dened by constraints yS ≤ xi ∀i ∈ S, ∀S ∈ S,sgn(aS) = +1 yS ≥X i ∈S xi− (|S| −1) ∀S ∈ S,sgn(aS) = −1

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