Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

Bao, Xiaowei; Sahinidis, Nikolaos V.; Tawarmalani, Mohit

doi:10.1007/s10107-011-0462-2

Cited by 119 publications

(106 citation statements)

References 42 publications

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“…The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model. It is standard to use termwise convex/concave envelopes [11,91] to relax MIQCQP, but many tighter relaxations have been developed based on: polyhedral facets of edge-concave multivariable term aggregations [17,26,34,94,95,96,99,111,130,131,132], eigenvector projections [38,106,113,122], piecewise-linear underestimators [29,65,66,73,93,98,99,100,101,107,119,139], outer approximation of convex terms [32,48,47], and semidefinite programming (SDP) relaxations [16,25,35,122,121]. GloMIQO incorporates several of these advanced relaxations.…”

Section: Literature Reviewmentioning

confidence: 99%

GloMIQO: Global mixed-integer quadratic optimizer

2012

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This paper introduces the Global Mixed-Integer Quadratic Optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2,

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Section: Literature Reviewmentioning

confidence: 99%

GloMIQO: Global mixed-integer quadratic optimizer

2012

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“…In this paper, we also allow conic convex constraints on the variables, so our problem of interest is a conic QCQP. Recent works [3,8,9,21,25,43,49] on the QCQP have developed algorithms of various kinds for solving such a program. Of particular relevance to our work herein are the most recent papers [3,21,25] that lift a QCQP satisfying a boundedness assumption to an equivalent completely positive program [23], which is a problem of topical interest.…”

Section: Introductionmentioning

confidence: 99%

On conic QPCCs, conic QCQPs and completely positive programs

2015

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This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a conic quadratically constrained quadratic program (QCQP), a conic quadratic program with complementarity constraints (QPCC), and rank constrained semidefinite programs. Our results do not make any boundedness assumptions on the feasible regions of the various problems considered. The first stage in the reformulation is to cast the problem as a conic QCQP with just one nonconvex constraint q(x) ≤ 0, where q(x) is nonnegative over the (convex) conic and linear constraints, so the problem fails the Slater constraint qualification. A conic QPCC has such a structure; we prove the converse, namely that any conic QCQP satisfying a constraint qualification can be expressed as an equivalent conic QPCC. of the reformulation lifts the problem to a completely positive program, and exploits and generalizes a result of Burer. We also show that a Frank-Wolfe type result holds for a subclass of this class of conic QCQPs. Further, we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper.Mathematics Subject Classification Primary 90C20; Secondary 65K05 · 90C11 · 90C22 · 90C25 · 90C26 · 90C33 · 90C46

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“…So, in a certain sense, solving (1) is equivalent to characterizing C (F ). In fact, many existing techniques for QCQP can be interpreted as providing tractable relaxations of C (F ); see [3,4]. Characterizing C (F ) in a tractable manner is difficult; if it were easy, then we could solve (1) easily.…”

Section: Introductionmentioning

confidence: 99%

Representing quadratically constrained quadratic programs as generalized copositive programs

Burer

Dong

2012

Operations Research Letters

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We show that any nonconvex quadratically constrained quadratic program (QCQP) can be represented as a generalized copositive program. In fact, we provide two representations. The first is based on the concept of completely positive (CP) matrices over second order cones, while the second is based on CP matrices over the positive semidefinte cone. Our analysis assumes that the feasible region of the QCQP is nonempty and bounded.

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Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

Cited by 119 publications

References 42 publications

GloMIQO: Global mixed-integer quadratic optimizer

GloMIQO: Global mixed-integer quadratic optimizer

On conic QPCCs, conic QCQPs and completely positive programs

Representing quadratically constrained quadratic programs as generalized copositive programs

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