Simultaneous Convexification of Bilinear Functions over Polytopes with Application to Network Interdiction

Davarnia, Danial; Richard, Jean‐Philippe P.; Tawarmalani, Mohit

doi:10.1137/16m1066166

Cited by 15 publications

(18 citation statements)

References 35 publications

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“…There are relatively lesser number of results on convexification of sets [38,28,29,36]. Theorem 1 generalizes results presented in [37,15,21] and is related to results presented in [9].…”

Section: Second Order Cone Representable Relaxation Of Bbpmentioning

confidence: 53%

New SOCP relaxation and branching rule for bipartite bilinear programs

2018

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A bipartite bilinear program (BBP) is a quadratically constrained quadratic optimization problem where the variables can be partitioned into two sets such that fixing the variables in any one of the sets results in a linear program. We propose a new second order cone representable (SOCP) relaxation for BBP, which we show is stronger than the standard SDP relaxation intersected with the boolean quadratic polytope. We then propose a new branching rule inspired by the construction of the SOCP relaxation. We describe a new application of BBP called as the finite element model updating problem, which is a fundamental problem in structural engineering. Our computational experiments on this problem class show that the new branching rule together with an polyhedral outer approximation of the SOCP relaxation outperforms a state-of-the-art commercial global solver in obtaining dual bounds.

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“…There are relatively lesser number of results on convexification of sets [38,28,29,36]. Theorem 1 generalizes results presented in [37,15,21] and is related to results presented in [9].…”

Section: Second Order Cone Representable Relaxation Of Bbpmentioning

confidence: 53%

New SOCP relaxation and branching rule for bipartite bilinear programs

2018

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“…A.1 Proof of Part (iii) of Proposition 2 Proposition 6. conv U row (n 1 ,n 2 ) (l, u) is described by the inequalities (18), (19), (20), and (21).…”

Section: A Omitted Proofsmentioning

confidence: 99%

“…Proposition 9. conv U row+ (n 1 ,n 2 ) (l, u, L, U ) is described by the inequalities (31), (32), (20), and (21).…”

Section: A2 Proof Of Part (Ii) Propositionmentioning

confidence: 99%

“…However, most of the work in this direction in the global optimization area has been focused on convexification of functions (i.e., finding convex and concave envelopes), see for example [4,46,33,10,36,9,8,38,21,48,45,37,52,53,16,19,1,26]. There are relatively lesser number of results on convexification of sets [51,41,42,50,24,30,44,20,32,17,40,23]. It is well-known that it is possible to obtain tighter convex relaxations when convexifing a set directly rather than using convex envelopes of functions describing the set.…”

Section: Introduction 1motivationmentioning

confidence: 99%

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A study of rank-one sets with linear side constraints and application to the pooling problem

Dey,

Kocuk,

Santana

2019

Preprint

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We study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral or second-order cone representable. In all these cases, we also show that a linear objective can be optimized in polynomial time over these sets. Towards the application side, we show how these sets relate to commonly occurring substructures of a general quadratically constrained quadratic program. To further illustrate the benefit of studying quadratically constrained quadratic programs from a rank-1 perspective, we propose new rank-1 formulations for the generalized pooling problem and use our convexification results to obtain several new convex relaxations for the pooling problem. Finally, we run a comprehensive set of computational experiments and show that our convexification results together with discretization significantly help in improving dual bounds for the generalized pooling problem.

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“…While there have been a lot of work in function convexification (see for instance [3,49,5,46,35,10,38,6,8,7,41,18,47,45,39,55,56,36,12,16,1,27,51]) it is well-known that it does not necessarily yield the convex hull of the set {x | f (x) = b}. To the best of our knowledge, there have been much less work on explicit convexification of sets: [54,42,43,53,25,32,44,17,34,13].…”

Section: Introductionmentioning

confidence: 99%

The convex hull of a quadratic constraint over a polytope

Santana¹,

Dey²

2018

Preprint

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A quadratically constrained quadratic program (QCQP) is an optimization problem in which the objective function is a quadratic function and the feasible region is defined by quadratic constraints. Solving non-convex QCQP to global optimality is a well-known NP-hard problem and a traditional approach is to use convex relaxations and branch-andbound algorithms. This paper makes a contribution in this direction by showing that the exact convex hull of a general quadratic equation intersected with any bounded polyhedron is second-order cone representable. We present a simple constructive proof of this result.

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Simultaneous Convexification of Bilinear Functions over Polytopes with Application to Network Interdiction

Cited by 15 publications

References 35 publications

New SOCP relaxation and branching rule for bipartite bilinear programs

New SOCP relaxation and branching rule for bipartite bilinear programs

A study of rank-one sets with linear side constraints and application to the pooling problem

The convex hull of a quadratic constraint over a polytope

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