Maximal Quadratic-Free Sets

Muñoz, Gonzalo; Serrano, Felipe

doi:10.1007/978-3-030-45771-6_24

Cited by 15 publications

(5 citation statements)

References 36 publications

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“…The quality of the resulting cut highly depends on which S-free set is used. The tightest possible intersection cuts are obtained by using maximal S-free sets as proposed by Muñoz and Serrano [79].…”

Section: Intersection Cuts For Quadratic Constraintsmentioning

confidence: 99%

The SCIP Optimization Suite 8.0

Bestuzheva¹,

Besançon²,

Chen³

et al. 2021

Preprint

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The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 8.0 of the SCIP Optimization Suite. Major updates in SCIP include improvements in symmetry handling and decomposition algorithms, new cutting planes, a new plugin type for cut selection, and a complete rework of the way nonlinear constraints are handled. Additionally, SCIP 8.0 now supports interfaces for Julia as well as Matlab. Further, UG now includes a unified framework to parallelize all solvers, a utility to analyze computational experiments has been added to GCG, dual solutions can be postsolved by PaPILO, new heuristics and presolving methods were added to SCIP-SDP, and additional problem classes and major performance improvements are available in SCIP-Jack. Keywords Constraint integer programming • linear programming • mixed-integer linear programming • mixed-integer nonlinear programming • optimization solver • branch-andcut • branch-and-price • column generation • parallelization • mixed-integer semidefinite programming Mathematics Subject Classification 90C05 • 90C10 • 90C11 • 90C30 • 90C90 • 65Y05 * Extended author information is available at the end of the paper.

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Section: Intersection Cuts For Quadratic Constraintsmentioning

confidence: 99%

The SCIP Optimization Suite 8.0

Bestuzheva¹,

Besançon²,

Chen³

et al. 2021

Preprint

Self Cite

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“…Further, [23] demonstrates that cuts obtained from one-row relaxations of QCQPs can be useful computationally. The paradigm of intersection cuts has also been explored to generate cuts for single-constraint QCQPs [40,16]. Due to lack of space, we refrain from describing here the vast literature on convexification techniques for QCQPs and instead refer interested readers to [18,52] and the references therein.…”

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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“…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

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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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Maximal Quadratic-Free Sets

Cited by 15 publications

References 36 publications

The SCIP Optimization Suite 8.0

The SCIP Optimization Suite 8.0

Lifting convex inequalities for bipartite bilinear programs

On obtaining the convex hull of quadratic inequalities via aggregations

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