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
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,
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!
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
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.
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.
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.
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,
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
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
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
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
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
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
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
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.
SL complete description
Multilinear 0-1 optimization minX S∈S aS Y i ∈S xi+ l (x ) s. t. xi ∈ {0, 1} i =1, . . . , nStandard 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.
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.
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.
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.
SL complete description: signed case
Theorem 4Given 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