2x2 convexifications for convex quadratic optimization with indicator variables

Han, Shaoning; Gómez, Andrés; Atamtürk, Alper

doi:10.48550/arxiv.2004.07448

Cited by 11 publications

(19 citation statements)

References 33 publications

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“…Set X 2×2 corresponds (after scaling) to a generic strictly convex quadratic function of two variables; conic quadratic disjunctive programming representations of cl conv(X 2×2 ) have been used in the literature [4,20], explicit representations of cl conv (X 2×2 ∩ {(x, z, t) : x ≥ 0}) have been given [8,24], and descriptions of the rank-one case d 1 d 2 = 1 were given in [6], but an explicit representation of cl conv(X 2×2 ) with free variables has not been established in the literature.…”

Section: Convexification In An Extended Spacementioning

confidence: 99%

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On the convex hull of convex quadratic optimization problems with indicators

Wei¹,

Atamtürk²,

Gómez³

et al. 2022

Preprint

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We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are "finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.

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Section: Convexification In An Extended Spacementioning

confidence: 99%

“…2 can be added to the formulation. Alternative generalizations of perspective relaxation that have been considered in the literature include exploiting substructures based on Γ i where non-zeros are 2x2 matrices [4,5,8,20,24,26] or tridiagonal [28].…”

Section: Introductionmentioning

confidence: 99%

On the convex hull of convex quadratic optimization problems with indicators

Wei¹,

Atamtürk²,

Gómez³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, a class of even tighter relaxations were developed by Atamtürk and Gomez [2], Han et al [34]. As they were developed by considering multiple binary variables simultaneously and therefore do not generalize readily to the low-rank case (where we often have one low-rank matrix), we do not discuss (or generalize) them here.…”

Section: Motivating Examplementioning

confidence: 99%

A new perspective on low-rank optimization

Bertsimas¹,

Cory-Wright²,

Pauphilet³

2021

Preprint

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A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function -the matrix analog of the perspective function -and characterize explicitly the convex hull of epigraphs of convex quadratic, matrix exponential, and matrix power functions under low-rank constraints. Further, we exploit these characterizations to develop strong relaxations for a variety of low-rank problems including reduced rank regression, non-negative matrix factorization, and factor analysis. We establish that these relaxations can be modeled via semidefinite and matrix power cone constraints, and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization, and leads to new relaxations for a broad class of problems.

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“…This convex hull characterization has led to the development of several techniques for problem (1) with general Q, including cutting plane methods [26,25], strong MISOCP formuations [1,32], approximation algorithms [56], specialized branching methods [34], and presolving methods [5]. Recently, problem (1) has been studied under other structural assumptions, including: quadratic terms involving two variables only [2,3,27,33,37], rank-one quadratic terms [4,6,51,50], and quadratic terms with Stieltjes matrices [7]. If the matrix can be factorized as Q = Q 0 Q 0 where Q 0 is sparse (but Q is dense), then problem (1) can be solved (under appropriate conditions) in polynomial time [21].…”

Section: Introductionmentioning

confidence: 99%

A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

Liu¹,

Fattahi²,

Gómez³

et al. 2021

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In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This enables us to construct a compact extended formulation for the closure of the convex hull of the epigraph of the mixed-integer convex problem. Furthermore, we propose a novel decomposition method for general (sparse) Q, which leverages the efficient algorithm for the path case. Our computational experiments demonstrate the effectiveness of the proposed method compared to stateof-the-art mixed-integer optimization solvers.

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2x2 convexifications for convex quadratic optimization with indicator variables

Cited by 11 publications

References 33 publications

On the convex hull of convex quadratic optimization problems with indicators

On the convex hull of convex quadratic optimization problems with indicators

A new perspective on low-rank optimization

A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

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