The Convex Hull of a Quadratic Constraint over a Polytope

Santana, Asteroide; Dey, Santanu S.

doi:10.1137/19m1277333

Cited by 18 publications

(6 citation statements)

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“…where a key challenge is to first ascertain that the supremum is finite. When α * is finite, it is clear that choosing any α ≥ α * in (1) yields a valid inequality for S. Problem (2) can be analyzed using the following facts: (1) for any fixed value of x 3 , we can always assume that an extreme point is the optimal solution, as the objective is to maximize a convex function, and (2) the extreme points of the set where x 3 is fixed to a value within its bounds are well-understood [52]. This suggests that one can inspect all different values of x 3 to establish that the supremum is finite.…”

Section: Resultsmentioning

confidence: 99%

“…Generating valid inequalities for single row relaxations (together with bounds and integrality restrictions), i.e., for knapsack constraints, was the first, and arguably the most important step in the development of computationally useful cutting-planes in MILP. Motivated by this observation, various cutting-planes and convexification techniques for sets defined by a single non-convex quadratic constraint together with bounds have recently been investigated; see [20,53,2] for classes of valid inequalities for single constraint QCQPs and [23,52] for convex hull results for such sets. The paper [45] studies a set similar to the one we study, albeit with integer variables.…”

Section: Goal Of This Papermentioning

confidence: 99%

“…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%

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Lifting convex inequalities for bipartite bilinear programs

Dey

Richard

2021

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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: Resultsmentioning

confidence: 99%

Section: Goal Of This Papermentioning

confidence: 99%

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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

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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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“…The closed convex hull is shown to be SOC-representable under certain conditions. More recently, Santana and Dey [53] showed that the convex hull of the intersection of a polytope and a single nonconvex quadratic constraints is SOC-representable.…”

Section: Remark 11mentioning

confidence: 99%

Exactness in SDP relaxations of QCQPs: Theory and applications

Kılınç-Karzan,

Wang

2021

Preprint

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Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Such problems arise naturally in many areas of operations research, computer science, and engineering. Although QCQPs are NP-hard to solve in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. In this tutorial, we will study the SDP relaxation for general QCQPs, present various exactness concepts related to this relaxation and discuss conditions guaranteeing such SDP exactness. In particular, we will define and examine three notions of SDP exactness: (i) objective value exactness-the condition that the optimal value of the QCQP and the optimal value of its SDP relaxation coincide, (ii) convex hull exactness-the condition that the convex hull of the QCQP epigraph coincides with the (projected) SDP epigraph, and (iii) the rank-one generated (ROG) property-the condition that a particular conic subset of the positive semidefinite matrices related to a given QCQP is generated by its rank-one matrices. Our analysis for objective value exactness and convex hull exactness stems from a geometric treatment of the projected SDP relaxation and crucially considers how the objective function interacts with the constraints. The ROG property complements these results by offering a sufficient condition for both objective value exactness and convex hull exactness which is oblivious to the objective function. By analyzing the geometry of the associated sets, we will give a variety of sufficient conditions for these exactness conditions and discuss settings where these sufficient conditions are additionally necessary. Throughout, we will highlight implications of our results for a number of example applications.

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The Convex Hull of a Quadratic Constraint over a Polytope

Cited by 18 publications

References 50 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

Exactness in SDP relaxations of QCQPs: Theory and applications

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