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Linear and quadratic reformulations of nonlinearoptimization problems in binary variables
Elisabeth Rodrıguez-HeckRWTH Aachen University, Chair of Operations Research
Results of PhD thesisadvised by Yves Crama, University of Liege
G-Scop, Grenoble, February 7, 2019
Introduction
Pseudo-Boolean optimization
General problem: pseudo-Boolean optimizationGiven a pseudo-Boolean function f : 0, 1n → R
minx∈0,1n
f (x).
Theorem (Hammer, Rosenberg, & Rudeanu, 1963)Every pseudo-Boolean function f : 0, 1n → R admits a uniquemultilinear expression.
I Given f , finding its unique multilinear representation can becostly! (Size of the input: O(2n))
p.2
Pseudo-Boolean optimization
General problem: pseudo-Boolean optimizationGiven a pseudo-Boolean function f : 0, 1n → R
minx∈0,1n
f (x).
Theorem (Hammer, Rosenberg, & Rudeanu, 1963)Every pseudo-Boolean function f : 0, 1n → R admits a uniquemultilinear expression.
I Given f , finding its unique multilinear representation can becostly! (Size of the input: O(2n))
p.2
Pseudo-Boolean optimization
General problem: pseudo-Boolean optimizationGiven a pseudo-Boolean function f : 0, 1n → R
minx∈0,1n
f (x).
Theorem (Hammer, Rosenberg, & Rudeanu, 1963)Every pseudo-Boolean function f : 0, 1n → R admits a uniquemultilinear expression.
I Given f , finding its unique multilinear representation can becostly! (Size of the input: O(2n))
p.2
Multilinear 0–1 optimization
Assumption: f given as a multilinear polynomialSet of monomials S ⊆ 2[n], aS 6= 0 for S ∈ S.
min∑S∈S
aS∏i∈S
xi
s. t. xi ∈ 0, 1, for i = 1, . . . , n
Example:
f (x1, x2, x3) = 9x1x2x3 + 8x1x2 − 6x2x3 − 2x1
p.3
Application in computer vision: image restoration
Input: blurred image Output: restored image
Image from the Corel database.
p.4
Applications
I Location problemsI Joint supply chain design and inventory managementI Statistical mechanicsI Quantum computingI ...
p.5
Linear and quadratic reformulations
Multilinear 0-1 optimizationminxi
∑S∈S aS
∏i∈S xi
Linear problemminxi ,yj l(x , y)
additional y variablesconstraints
Quadratic problemminxi ,yj q(x , y)
additional y variableswithout constraints (choice)
reformulate as reformulate as
p.6
Overview
Introduction:Multilinear 0-1 optimization
Part I: Linear re-formulations
Part II: Quadraticreformulations
Part III: Computa-tional experiments
Conclusions andPerspectives
Overview
Introduction:Multilinear 0-1 optimization
Part I: Linear re-formulations
Part II: Quadraticreformulations
Part III: Computa-tional experiments
Conclusions andPerspectives
Introduction to linearizations
The standard linearization (SL)Nonlinear problem
min∑S∈S
aS∏i∈S
xi +n∑
i=1ci xi
Linearized problem
min∑S∈S
aSyS +n∑
i=1ci xi
Standard Linearization (Fortet, 1959; Glover & Woolsey, 1973)
yS =∏i∈S
xi
When xi ∈ 0, 1 it is equivalent to say:
yS ≤ xi ∀i ∈ S,∀S ∈ S (1)yS ≥
∑i∈S
xi − (|S| − 1) ∀S ∈ S (2)
p.7
The standard linearization (SL)Nonlinear problem
min∑S∈S
aS∏i∈S
xi +n∑
i=1ci xi
Linearized problem
min∑S∈S
aSyS +n∑
i=1ci xi
Standard Linearization (Fortet, 1959; Glover & Woolsey, 1973)
yS =∏i∈S
xi
When xi ∈ 0, 1 it is equivalent to say:
yS ≤ xi ∀i ∈ S,∀S ∈ S (1)yS ≥
∑i∈S
xi − (|S| − 1) ∀S ∈ S (2)
p.7
SL drawback: The continuous relaxation given by theSL is very weak!
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Image from A. Schrijver’s book cover:Theory of linear and integer programming.p.8
When do the SL inequalities provide a perfectformulation?
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p.9
Hypergraph associated with a polynomialA multilinear polynomial can be associated to a hypergraphH = (V = 1, . . . , n,S).
min∑S∈S
aS∏i∈S
xi +∑i∈V
ci xi
Example:
f (x1, x2, x3) = 9x1x2x3 − 8x1x2 + 6x2x3 − 2x1
• •
•
x1 x2
x3
p.10
Characterization for the unsigned case
Theorem 1: (Buchheim, Crama, & Rodrıguez-Heck, 2019)Given a hypergraph H, the following statements are equivalent:(a) The SL inequalities define an integer polytope.(b) The matrix of coefficients of the SL inequalities is balanced.(c) The hypergraph H is Berge-acyclic.
I Obtained independently and simultaneously by (Del Pia &Khajavirad, 2018).
I Theorem 1 is a Corollary of a more general result consideringthe signs of the coefficients.
I The balancedness condition can be checked in polynomialtime.
p.11
Characterization for the unsigned case
Theorem 1: (Buchheim, Crama, & Rodrıguez-Heck, 2019)Given a hypergraph H, the following statements are equivalent:(a) The SL inequalities define an integer polytope.(b) The matrix of coefficients of the SL inequalities is balanced.(c) The hypergraph H is Berge-acyclic.
I Obtained independently and simultaneously by (Del Pia &Khajavirad, 2018).
I Theorem 1 is a Corollary of a more general result consideringthe signs of the coefficients.
I The balancedness condition can be checked in polynomialtime.
p.11
Characterization for the unsigned case
Theorem 1: (Buchheim, Crama, & Rodrıguez-Heck, 2019)Given a hypergraph H, the following statements are equivalent:(a) The SL inequalities define an integer polytope.(b) The matrix of coefficients of the SL inequalities is balanced.(c) The hypergraph H is Berge-acyclic.
I Obtained independently and simultaneously by (Del Pia &Khajavirad, 2018).
I Theorem 1 is a Corollary of a more general result consideringthe signs of the coefficients.
I The balancedness condition can be checked in polynomialtime.
p.11
Characterization for the unsigned case
Theorem 1: (Buchheim, Crama, & Rodrıguez-Heck, 2019)Given a hypergraph H, the following statements are equivalent:(a) The SL inequalities define an integer polytope.(b) The matrix of coefficients of the SL inequalities is balanced.(c) The hypergraph H is Berge-acyclic.
I Obtained independently and simultaneously by (Del Pia &Khajavirad, 2018).
I Theorem 1 is a Corollary of a more general result consideringthe signs of the coefficients.
I The balancedness condition can be checked in polynomialtime.
p.11
A class of valid inequalities for multilinear 0–1optimization problems
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p.12
The 2-link inequalities
Definition (Crama & Rodrıguez-Heck, 2017)For S,T ∈ S and yS , yT such that yS =
∏i∈S xi , yT =
∏i∈T xi ,
the 2-link associated with (S,T ) is the linear inequality
yS ≤ yT −∑
i∈T\S xi + |T\S|
Interpretation
S TyS = 1⇒ ∀i ∈ S, xi = 1
yT = 0 and yS = 1⇒ ∃j ∈ T\S, xj = 0
p.13
A complete description for the case of two monomials
Theorem 2: (Crama & Rodrıguez-Heck, 2017)For the case of two nonlinear monomials, P∗SL = P2links
SL , i.e., thestandard linearization and the 2-links provide a completedescription of P∗SL.
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p.14
A complete description for the case of two monomials
Theorem 2: (Crama & Rodrıguez-Heck, 2017)For the case of two nonlinear monomials, P∗SL = P2links
SL , i.e., thestandard linearization and the 2-links provide a completedescription of P∗SL.
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p.14
Proof idea: yS∩T =∏
i∈S∩T xi , yS = yS∩T∏
i∈S\T xi , yT = yS∩T∏
i∈T\S xi
I Consider the extended formulation (with variables in [0, 1])
yS∩T ≤ xi , ∀i ∈ S ∩ T , (3)
yS∩T ≥∑
i∈S∩T
xi − (|S ∩ T | − 1), (4)
yS ≤ yS∩T , (5)yS ≤ xi , ∀i ∈ S\T , (6)
yS ≥∑
i∈S\T
xi + yS∩T − |S\T |, (7)
yT ≤ yS∩T , (8)yT ≤ xi , ∀i ∈ T\S, (9)
yT ≥∑
i∈T\S
xi + yS∩T − |T\S|, (10)
I Notice that the two polytopes P0 and P1 obtained by fixing variableyS∩T to 0 and 1, resp., are integral.
I Compute conv(P0 ∪ P1) using (Balas, 1974) and see that it isP2links
SL .p.15
Overview
Introduction:Multilinear 0-1 optimization
Part I: Linear re-formulations
Part II: Quadraticreformulations
Part III: Computa-tional experiments
Conclusions andPerspectives
Introduction to quadratizations
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?I Small number of auxiliary variables (compact).I Small number of positive quadratic terms (xixj , xiyj . . . ) (empirical
distance from submodularity).I Set of quadratic terms with specific underlying graphs.I ...
p.16
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?
I Small number of auxiliary variables (compact).I Small number of positive quadratic terms (xixj , xiyj . . . ) (empirical
distance from submodularity).I Set of quadratic terms with specific underlying graphs.I ...
p.16
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?I Small number of auxiliary variables (compact).
I Small number of positive quadratic terms (xixj , xiyj . . . ) (empiricaldistance from submodularity).
I Set of quadratic terms with specific underlying graphs.I ...
p.16
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?I Small number of auxiliary variables (compact).I Small number of positive quadratic terms (xixj , xiyj . . . ) (empirical
distance from submodularity).
I Set of quadratic terms with specific underlying graphs.I ...
p.16
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?I Small number of auxiliary variables (compact).I Small number of positive quadratic terms (xixj , xiyj . . . ) (empirical
distance from submodularity).I Set of quadratic terms with specific underlying graphs.
I ...
p.16
Quadratization: definition and desirable properties
Definition (Anthony, Boros, Crama, & Gruber, 2017)Given a pseudo-Boolean function f (x) where x ∈ 0, 1n, aquadratization g(x , y) is a function satisfyingI g is quadraticI g(x , y) depends on the original variables x and on m auxiliary
variables yI satisfies
f (x) = ming(x , y) : y ∈ 0, 1m ∀x ∈ 0, 1n.
Which quadratizations are “good”?I Small number of auxiliary variables (compact).I Small number of positive quadratic terms (xixj , xiyj . . . ) (empirical
distance from submodularity).I Set of quadratic terms with specific underlying graphs.I ...
p.16
Persistencies
Weak Persistency Theorem (Hammer, Hansen, & Simeone, 1984)Let (QP) be a quadratic optimization problem on x ∈ 0, 1n, and let (x , y) bean optimal solution of the continuous standard linearization of (QP)
min c0 +n∑
j=1
cjxj +∑
1≤i<j≤n
cijyij
s. t. yij ≥ xi + xj − 1 i , j = 1, . . . , n, i < jyij ≤ xi i , j = 1, . . . , n, i < jyij ≤ xj i , j = 1, . . . , n, i < j0 ≤ yij ≤ 1 i , j = 1, . . . , n, i < j0 ≤ xi ≤ 1 i = 1, . . . , n
such that xj = 1 for j ∈ O and xj = 0 for j ∈ Z . Then, for any minimizingvector x∗ of (QP) switching x∗j = 1 for j ∈ O and x∗j = 0 for j ∈ Z will alsoyield a minimum of f .
(See also survey (Boros & Hammer, 2002).)p.17
Persistencies
I The Weak Persistency Theorem is not the strongest form ofpersistency.
I There are ways to compute, in polynomial time, a maximal setof variables to fix, based on a network flow algorithm (Boros,Hammer, Sun, & Tavares, 2008).
I In computer vision, image restoration and related problems ofup to millions of variables are efficiently solved, thanks to theuse of persistencies.
p.18
Persistencies
I The Weak Persistency Theorem is not the strongest form ofpersistency.
I There are ways to compute, in polynomial time, a maximal setof variables to fix, based on a network flow algorithm (Boros,Hammer, Sun, & Tavares, 2008).
I In computer vision, image restoration and related problems ofup to millions of variables are efficiently solved, thanks to theuse of persistencies.
p.18
Persistencies
I The Weak Persistency Theorem is not the strongest form ofpersistency.
I There are ways to compute, in polynomial time, a maximal setof variables to fix, based on a network flow algorithm (Boros,Hammer, Sun, & Tavares, 2008).
I In computer vision, image restoration and related problems ofup to millions of variables are efficiently solved, thanks to theuse of persistencies.
p.18
Termwise quadratizations
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Check that, for every x ∈ 0, 1n,miny g(x , y) = −
∏ni=1 xi ., two cases:
1 If xi = 1 ∀i , then miny − y ,minimum value of −1 reached fory = 1.
2 If ∃i such that xi = 0, thenminy − Cy , where C ≤ 0,minimum value of 0 reached fory = 0.
p.19
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Check that, for every x ∈ 0, 1n,miny g(x , y) = −
∏ni=1 xi ., two cases:
1 If xi = 1 ∀i , then miny − y ,minimum value of −1 reached fory = 1.
2 If ∃i such that xi = 0, thenminy − Cy , where C ≤ 0,minimum value of 0 reached fory = 0.
p.19
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Check that, for every x ∈ 0, 1n,miny g(x , y) = −
∏ni=1 xi ., two cases:
1 If xi = 1 ∀i , then miny − y ,minimum value of −1 reached fory = 1.
2 If ∃i such that xi = 0, thenminy − Cy , where C ≤ 0,minimum value of 0 reached fory = 0.
p.19
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Check that, for every x ∈ 0, 1n,miny g(x , y) = −
∏ni=1 xi ., two cases:
1 If xi = 1 ∀i , then miny − y ,minimum value of −1 reached fory = 1.
2 If ∃i such that xi = 0, thenminy − Cy , where C ≤ 0,minimum value of 0 reached fory = 0.
p.19
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Check that, for every x ∈ 0, 1n,miny g(x , y) = −
∏ni=1 xi ., two cases:
1 If xi = 1 ∀i , then miny − y ,minimum value of −1 reached fory = 1.
2 If ∃i such that xi = 0, thenminy − Cy , where C ≤ 0,minimum value of 0 reached fory = 0.
p.19
Termwise quadratizations
Main ideaQuadratize monomial by monomial using disjoint sets of auxiliary variables.
f (x) = −35x1x2x3x4x5 + 50x1x2x3x4 − 10x1x2x4x5 + 5x2x3x4 + 5x4x5 − 20x1
Negative monomial(Kolmogorov & Zabih, 2004; Freedman& Drineas, 2005)
−n∏
i=1
xi = miny∈0,1
−y(n∑
i=1
xi − (n − 1))
I One variable is sufficient.I No positive quadratic terms.
Positive monomial(Ishikawa, 2011)
n∏i=1
xi = miny∈0,1k
k∑i=1
yi (ci,n(−|x |+ 2i)− 1)
+|x |(|x | − 1)
2,
I Number of variables: k = b n−12 c.
I(n
2
)positive quadratic terms.
p.19
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .
I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.
I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented asa sum of log(n) powers of 2.
I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.
I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.I Use y variables to express which powers of 2 are in the sum.
I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.
p.20
Upper bound for the positive monomial: dlog(n)e − 1
Theorem 3 (simplified) (Boros, Crama, & Rodrıguez-Heck,2018)Assume that n = 2` and let |x | =
∑ni=1 xi be the Hamming weight of
x ∈ 0, 1n. Then,
g(x , y) = 12 (|x | −
`−1∑i=1
2iyi )(|x | −`−1∑i=1
2iyi − 1)
is a quadratization of the positive monomial Pn(x) =∏n
i=1 xi usingdlog(n)e − 1 auxiliary variables.
Proof idea: Check that, for every x ∈ 0, 1n, miny g(x , y) =∏n
i=1 xi .I The quadratization depends on |x |, which takes values between 0 and n.I Case 1 (|x | ≤ n− 1): Integers between 0 and n− 1 can be represented as
a sum of log(n) powers of 2.I Use y variables to express which powers of 2 are in the sum.I For |x | ≤ n − 1, one factor to reach the minimum value of zero for odd|x | and the other factor for even |x |.
I Case 2 (|x | = n): Similarly, we can show miny g(x , y) = 1.p.20
Lower bound for the positive monomial
Theorem 4 (Boros, Crama, & Rodrıguez-Heck, 2018)If g(x , y) is a quadratization of the positive monomial Pn(x) =
∏ni=1 xi using m
variables, thenm ≥ dlog(n)e − 1
Proof idea:I Consider r(x) =
∏y∈0,1m g(x , y).
I deg(r) ≤ 2 · 2m.I deg(r) ≥ n, because r(x) = αPn(x) where α > 0 (unicity of the
multilinear representation). More precisely,I If |x | < n, there exists y ∈ 0, 1m such that g(x , y) = 0.I If |x | = n, g(x , y) ≥ 1 for all y ∈ 0, 1m.
p.21
Lower bound for the positive monomial
Theorem 4 (Boros, Crama, & Rodrıguez-Heck, 2018)If g(x , y) is a quadratization of the positive monomial Pn(x) =
∏ni=1 xi using m
variables, thenm ≥ dlog(n)e − 1
Proof idea:
I Consider r(x) =∏
y∈0,1m g(x , y).
I deg(r) ≤ 2 · 2m.I deg(r) ≥ n, because r(x) = αPn(x) where α > 0 (unicity of the
multilinear representation). More precisely,I If |x | < n, there exists y ∈ 0, 1m such that g(x , y) = 0.I If |x | = n, g(x , y) ≥ 1 for all y ∈ 0, 1m.
p.21
Lower bound for the positive monomial
Theorem 4 (Boros, Crama, & Rodrıguez-Heck, 2018)If g(x , y) is a quadratization of the positive monomial Pn(x) =
∏ni=1 xi using m
variables, thenm ≥ dlog(n)e − 1
Proof idea:I Consider r(x) =
∏y∈0,1m g(x , y).
I deg(r) ≤ 2 · 2m.I deg(r) ≥ n, because r(x) = αPn(x) where α > 0 (unicity of the
multilinear representation). More precisely,I If |x | < n, there exists y ∈ 0, 1m such that g(x , y) = 0.I If |x | = n, g(x , y) ≥ 1 for all y ∈ 0, 1m.
p.21
Lower bound for the positive monomial
Theorem 4 (Boros, Crama, & Rodrıguez-Heck, 2018)If g(x , y) is a quadratization of the positive monomial Pn(x) =
∏ni=1 xi using m
variables, thenm ≥ dlog(n)e − 1
Proof idea:I Consider r(x) =
∏y∈0,1m g(x , y).
I deg(r) ≤ 2 · 2m.
I deg(r) ≥ n, because r(x) = αPn(x) where α > 0 (unicity of themultilinear representation). More precisely,I If |x | < n, there exists y ∈ 0, 1m such that g(x , y) = 0.I If |x | = n, g(x , y) ≥ 1 for all y ∈ 0, 1m.
p.21
Lower bound for the positive monomial
Theorem 4 (Boros, Crama, & Rodrıguez-Heck, 2018)If g(x , y) is a quadratization of the positive monomial Pn(x) =
∏ni=1 xi using m
variables, thenm ≥ dlog(n)e − 1
Proof idea:I Consider r(x) =
∏y∈0,1m g(x , y).
I deg(r) ≤ 2 · 2m.I deg(r) ≥ n, because r(x) = αPn(x) where α > 0 (unicity of the
multilinear representation). More precisely,I If |x | < n, there exists y ∈ 0, 1m such that g(x , y) = 0.I If |x | = n, g(x , y) ≥ 1 for all y ∈ 0, 1m.
p.21
Results for more general functionsFunction Lower Bound Upper Bound
Zero until k Ω(2n2 ) for some function1 O(2
n2 ) 1
dlog(k)e − 1 for all functions
Symmetric Ω(√
n) for some function2 O(√
n) = 2d√
n + 1e
Exact k-out-of-n max(dlog(k)e, dlog(n − k)e)− 1 max(dlog(k)e, dlog(n − k)e)
At least k-out-of-n dlog(k)e − 1 max(dlog(k)e, dlog(n − k)e)
Positive monomial dlog(n)e − 1 dlog(n)e − 1
Parity dlog(n)e − 1 dlog(n)e − 1
Zero until k Symmetric
Exact k-out-of-n At least k-out-of-n Parity
Positive monomial
1see (Anthony et al., 2017)2see (Anthony, Boros, Crama, & Gruber, 2016)
p.22
Pairwise covers
Pairwise covers
Anthony, Boros, Crama and Gruber (2017)
Substituting common sets of variables
f (x) = −35x1x2x3x4x5 +50x1x2x3x4−10x1x2x4x5 +5x2x3x4 +5x4x5−20x1
could be replaced by
f (x) = −35y12y345 +50y12y34−10y12y45 +5x2y34 +5x4x5−20x1 +P(x , y)
where P(x , y) imposes y12 = x1x2, y345 = y34x5...
p.23
Heuristics for small Pairwise Covers
Three heuristics:I PC1: Separate first two variables from the rest.I PC2: Most “popular” intersections first.I PC3: Most “popular” pairs of variables first.
Main idea: identifying subterms that appear as subsets of one ormore monomials in the input monomial set S.
p.24
Overview
Introduction:Multilinear 0-1 optimization
Part I: Linear re-formulations
Part II: Quadraticreformulations
Part III: Computa-tional experiments
Conclusions andPerspectives
Comparing linear and quadraticreformulations
Methods
MethodsLinearizations Quadratizations
2-links Pairw. cov. TermwiseSL SL-2L PC1 PC2 PC3 Ishikawa logn-1
p.25
Objectives and scope
Objectives1 Compare resolution times of linear and quadratic
reformulations when relying on a commercial solver.2 Are the results consistent for different types of instances?
The objective is not to compete with existing literature.
Dependence on the underlying solver: CPLEX 12.7Drawbacks:
7 Linear solver possibly more powerful than quadratic solver.7 “Blackbox”: little control over the actual resolution.7 Cannot exploit interesting properties like persistencies.
Advantages:3 Commercial widely used solver.3 Avoid comparing commercial against “home-made” software.
p.26
Objectives and scope
Objectives1 Compare resolution times of linear and quadratic
reformulations when relying on a commercial solver.2 Are the results consistent for different types of instances?
The objective is not to compete with existing literature.
Dependence on the underlying solver: CPLEX 12.7Drawbacks:
7 Linear solver possibly more powerful than quadratic solver.7 “Blackbox”: little control over the actual resolution.7 Cannot exploit interesting properties like persistencies.
Advantages:3 Commercial widely used solver.3 Avoid comparing commercial against “home-made” software.
p.26
Application in computer vision: image restoration
Input: blurred image Output: restored image
Image from the Corel database.
p.27
Instances: Vision
0 0 0 0 0 00 0 1 1 0 10 1 1 1 1 00 1 1 0 1 00 0 1 1 0 01 0 0 0 0 0
Image restoration
0 0 0 0 0 00 0 1 1 0 00 1 1 1 1 00 1 1 1 1 00 0 1 1 0 00 0 0 0 0 0
Base images:I top left rect. (tl)I centre rect. (cr)I cross (cx)
Perturbations:I none (n)I low (l)I high (h)
Up to n = 900 variables and m = 6788 terms
p.28
Vision: all methods 15× 15 (n = 225, m = 1598)
tl-stl-
l-1tl-
l-2tl-
h-1tl-
h-2 cr-scr-
l-1cr-
l-2cr-
h-1cr-
h-2 cx-scx-
l-1cx-
l-2cx-
h-1cx-
h-20
100
200
300
400
Instances
Tim
e(s
)
SL-2LSL
PC1PC2PC3
Ishikawalogn-1
p.29
Vision: best methods 15× 15 (n = 225, m = 1598)
tl-stl-
l-1tl-
l-2tl-
h-1tl-
h-2 cr-scr-
l-1cr-
l-2cr-
h-1cr-
h-2 cx-scx-
l-1cx-
l-2cx-
h-1cx-
h-20
20
40
60
80
100
120
140
Instances
Tim
e(s
)SL-2L
SLPC1PC2PC3
p.30
Vision: Quadratization properties
Pairwise covers Termwise
Number of y variables less moreNumber of positive less morequadratic termsMonomial interactions 3 7
p.31
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Dynamical glass transition phenomenon
More on YouTube: supercooled water, Lenape High School - science skills class.Scientific articles: (Bernasconi, 1987; Liers, Marinari, Pagacz, Ricci-Tersenghi,
& Schmitz, 2010).Instances downloaded from
http://polip.zib.de/autocorrelated sequences/.
p.32
Autocorrelated sequences
At least one method finished in 1h for instances in bold: 8/40 (!).Id n m Id n m20 5 20 207 45 5 45 50720 10 20 833 45 11 45 2 81325 6 25 407 45 23 45 10 77625 13 25 1 782 45 34 45 18 34825 19 25 3 040 45 45 45 21 99325 25 25 3 677 50 6 50 88230 4 30 223 50 13 50 4 45730 8 30 926 50 25 50 14 41230 15 30 2 944 50 38 50 25 44630 23 30 5 376 50 50 50 30 27130 30 30 6 412 55 6 55 97735 4 35 263 55 14 55 5 79035 9 35 1 381 55 28 55 19 89735 18 35 5 002 55 41 55 33 31835 26 35 8 347 55 55 55 40 40240 5 40 447 60 8 60 2 03640 10 40 2 053 60 15 60 7 29440 20 40 7 243 60 30 60 25 23040 30 40 12 690 60 45 60 43 68940 40 40 15 384 60 60 60 52 575
p.33
Autocorrelated sequences: preliminary results
Instance Resolution time (secs)
Id n m SL-2L SL SL-2L-Only SL-NoCuts
20 5 20 207 6.94 11.48 1.06 6.1320 10 20 833 112.27 91.75 47 65.8925 6 25 407 65.36 137.38 44.52 321.7730 4 30 223 4.24 15.7 13.47 349.8435 4 35 263 11.92 36.86 69.03 3104.4525 13 25 1782 1645.2 2567.17 685.74 2408.6930 8 30 926 2743.51 > 3600 > 3600 > 360040 5 40 447 1321.61 > 3600 > 3600 > 3600
Instance Resolution time (secs)
ID n m PC1 PC2 PC3 Ishikawa logn-1
20 5 20 207 10.58 5.05 4.27 37.47 35.3420 10 20 833 90.28 159.47 137.69 417.72 365.4725 6 25 407 106.67 80.17 121.03 629.66 466.9230 4 30 223 13.52 7.17 7.03 29.67 36.0835 4 35 263 24.13 13.25 11.2 49.77 54.1425 13 25 1782 2311.09 > 3600 > 3600 > 3600 > 360030 8 30 926 > 3600 > 3600 > 3600 > 3600 > 360040 5 40 447 > 3600 914.27 2053.97 > 3600 > 3600
p.34
Autocorr. seq.: Quadratization properties
Pairwise covers Termwise
Number of y variables less moreNumber of positive less morequadratic termsMonomial interactions 3 7
p.35
Instances: Random high degree
What happens with random polynomials?
Generated as in (Buchheim & Rinaldi, 2007).I Degree of each monomial chosen randomly (higher probability
for lower degrees).I The variables in monomials are randomly chosen, and the
coefficients.I Degrees vary between 7 and 17.
p.36
Random high degree: all methods
200-2
50-9-
1
200-2
50-9-
2
200-2
50-9-
3
200-2
50-9-
4
200-2
50-7-
5
200-3
00-8-
1
200-3
00-9-
2
200-3
00-15
-3
200-3
00-13
-4
200-3
00-12
-5
200-3
50-10
-1
200-3
50-10
-2
200-3
50-13
-3
200-3
50-9-
4
200-3
50-11
-5
200-4
00-14
-1
200-4
00-11
-2
200-4
00-11
-3
200-4
00-16
-4
200-4
00-12
-50
200
400
600
Instances
Tim
e(s
)
SLPC1PC2PC3
Ishikawalogn-1
p.37
Random high degree: best methods
200-2
50-9-
1
200-2
50-9-
2
200-2
50-9-
3
200-2
50-9-
4
200-2
50-7-
5
200-3
00-8-
1
200-3
00-9-
2
200-3
00-15
-3
200-3
00-13
-4
200-3
00-12
-5
200-3
50-10
-1
200-3
50-10
-2
200-3
50-13
-3
200-3
50-9-
4
200-3
50-11
-5
200-4
00-14
-1
200-4
00-11
-2
200-4
00-11
-3
200-4
00-16
-4
200-4
00-12
-50
5
10
15
20
25
Instances
Tim
e(s
)SL
PC1PC2PC3
p.38
Random high degree: Quadratization properties
Pairwise covers Termwise
Number of y variables more lessNumber of positive less morequadratic termsMonomial interactions 3 7
p.39
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:I Pairwise covers are faster than SL with default CPLEX branch
& cut.I SL with 2-links is faster than pairwise covers.
I For all instances, termwise quadratizations are slower thanpairwise covers.
I Minimizing the number of auxiliary variables is not the onlycriterion to consider.
p.40
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:
I Pairwise covers are faster than SL with default CPLEX branch& cut.
I SL with 2-links is faster than pairwise covers.I For all instances, termwise quadratizations are slower than
pairwise covers.I Minimizing the number of auxiliary variables is not the only
criterion to consider.
p.40
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:I Pairwise covers are faster than SL with default CPLEX branch
& cut.
I SL with 2-links is faster than pairwise covers.I For all instances, termwise quadratizations are slower than
pairwise covers.I Minimizing the number of auxiliary variables is not the only
criterion to consider.
p.40
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:I Pairwise covers are faster than SL with default CPLEX branch
& cut.I SL with 2-links is faster than pairwise covers.
I For all instances, termwise quadratizations are slower thanpairwise covers.
I Minimizing the number of auxiliary variables is not the onlycriterion to consider.
p.40
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:I Pairwise covers are faster than SL with default CPLEX branch
& cut.I SL with 2-links is faster than pairwise covers.
I For all instances, termwise quadratizations are slower thanpairwise covers.
I Minimizing the number of auxiliary variables is not the onlycriterion to consider.
p.40
Summary of results
I Linearizations are in general solved faster thanquadratizations, which is expected for CPLEX.
I For vision instances the behavior is different and rathersurprising:I Pairwise covers are faster than SL with default CPLEX branch
& cut.I SL with 2-links is faster than pairwise covers.
I For all instances, termwise quadratizations are slower thanpairwise covers.
I Minimizing the number of auxiliary variables is not the onlycriterion to consider.
p.40
Overview
Introduction:Multilinear 0-1 optimization
Part I: Linear re-formulations
Part II: Quadraticreformulations
Part III: Computa-tional experiments
Conclusions andPerspectives
Conclusions
Conclusions
Summary
I We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.
I We derived several theoretical results (strong linearformulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.
I Some properties to define good reformulationsI Exploiting structural properties (worked best for vision and
physics problems).I For quadratizations, understand number of positive quadratic
terms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Conclusions
SummaryI We considered linear and quadratic reformulations.I We derived several theoretical results (strong linear
formulations, study of properties of quadratizations, smallnumber of auxiliary variables...).
I We compared the reformulations computationally.I Some properties to define good reformulations
I Exploiting structural properties (worked best for vision andphysics problems).
I For quadratizations, understand number of positive quadraticterms influence.
p.41
Open questions
Theoretical
I Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?
I In general, understand better which properties define a“good” quadratization, theory and practice.
I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.
I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
Experiments
I Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
ExperimentsI Experiments are being re-tested using persistencies.
I Repeat experiments with other solvers (convexification,SDP...).
I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).
I ...
p.42
Open questions
TheoreticalI Which one is stronger: SL or SL of a quadratization?I In general, understand better which properties define a
“good” quadratization, theory and practice.I ...
ExperimentsI Experiments are being re-tested using persistencies.I Repeat experiments with other solvers (convexification,
SDP...).I ...
p.42
Open question: relaxing the multilinear assumption
Joint supply chain design &inventory management problem(Shen, Coullard, & Daskin, 2003;You & Grossmann, 2008)
A pseudo-Boolean formulation(with constraints)
min∑j∈J
fjxj +∑i∈I
dijyij+
Kj
√∑i∈I
µi yij + q√∑
i∈Iσ2
i yij
s.t.∑j∈J
yij = 1, ∀i ∈ I
yij ≤ xj , ∀i ∈ I,∀j ∈ Jxj ∈ 0, 1, yij ∈ 0, 1 ∀i ∈ I,∀j ∈ J
p.43
Open question: relaxing the multilinear assumption
Joint supply chain design &inventory management problem(Shen, Coullard, & Daskin, 2003;You & Grossmann, 2008)
A pseudo-Boolean formulation(with constraints)
min∑j∈J
fjxj +∑i∈I
dijyij+
Kj
√∑i∈I
µi yij + q√∑
i∈Iσ2
i yij
s.t.∑j∈J
yij = 1, ∀i ∈ I
yij ≤ xj , ∀i ∈ I,∀j ∈ Jxj ∈ 0, 1, yij ∈ 0, 1 ∀i ∈ I,∀j ∈ J
p.43
Thank you for your attention!rodriguez-heck@or.rwth-aachen.de
p.44
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