Lifting convex inequalities for bipartite bilinear programs

Gu, Xiaoyi; Dey, Santanu S.; Richard, Jean‐Philippe P.

doi:10.1007/s10107-021-01759-3

Cited by 3 publications

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“…To the best of our knowledge, this is the first study that derives lifted valid inequalities for general non-convex quadratic constraints with arbitrary number of variables. An extended abstract of this paper was accepted for publication in IPCO 2021 [30].…”

Section: Goal Of This Papermentioning

confidence: 99%

Lifting convex inequalities for bipartite bilinear programs

Dey

Richard

2021

Preprint

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The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We argue that this bilinear cover inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bipartite bilinear sets with box constraints.

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Section: Goal Of This Papermentioning

confidence: 99%

Lifting convex inequalities for bipartite bilinear programs

Dey

Richard

2021

Preprint

Self Cite

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“…The results presented in this paper contribute to the current literature on understanding the structure of the convex hull of simple sets described by quadratic constraints. Recently, the convex hull of one quadratic constraint intersected with a polytope, and other cutting-plane generation techniques for this set have been studied in [23,14,22,6,19,15]. The convex hull for two quadratic constraints and related sets have been studied in [28,10,17,13].…”

Section: Literature Reviewmentioning

confidence: 99%

On obtaining the convex hull of quadratic inequalities via aggregations

Dey¹,

Muñoz²,

Serrano³

2021

Preprint

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A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran [28] showed that the convex hull of the set is given by at most two aggregated inequalities. In this work, we study the case of a set described by three or more quadratic inequalities. We show that, under technical assumptions, the convex hull of a set described by three quadratic inequalities can be obtained via (potentially infinitely many) aggregated inequalities. We also show, through counterexamples, that it is unlikely to have a similar result if either the technical conditions are relaxed, or if we consider four or more inequalities.

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Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

Gu,

Dey,

Richard

2024

INFORMS Journal on Computing

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Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

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Lifting convex inequalities for bipartite bilinear programs

Cited by 3 publications

References 56 publications

Lifting convex inequalities for bipartite bilinear programs

Lifting convex inequalities for bipartite bilinear programs

On obtaining the convex hull of quadratic inequalities via aggregations

Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities

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