The MILP Road to MIQCP

Burer, Samuel; Saxena, Anureet

doi:10.1007/978-1-4614-1927-3_13

Cited by 47 publications

(40 citation statements)

References 53 publications

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“…We call an SOC constraint obtained in this fashion an "SOC-RLT" constraint. Subsequent to [19], SOC-RLT constraints have also appeared in [21] and [4]. In this paper, we study TRS with extra constraints, and in particular, we are interested in tight SDP relaxations that employ ordinary RLT and SOC-RLT constraints.…”

Section: Introductionmentioning

confidence: 99%

Second-Order-Cone Constraints for Extended Trust-Region Subproblems

2013

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The classical trust-region subproblem (TRS) minimizes a nonconvex quadratic objective over the unit ball. In this paper, we consider extensions of TRS having extra constraints. When two parallel cuts are added to TRS, we show that the resulting nonconvex problem has an exact representation as a semidefinite program with additional linear and second-order-cone constraints. For the case where an additional ellipsoidal constraint is added to TRS, resulting in the "two trust-region subproblem" (TTRS), we provide a new relaxation including second-order-cone constraints that strengthens the usual SDP relaxation.

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Section: Introductionmentioning

confidence: 99%

Second-Order-Cone Constraints for Extended Trust-Region Subproblems

2013

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“…Since then, disjunctive cuts have been studied extensively in mixed integer linear and nonlinear optimization [6,30,7,19,29,17,24,16]. Chvátal-Gomory, lift-and-project, mixed-integer rounding (MIR), and split cuts are all special types of disjunctive cuts.…”

Section: Introductionmentioning

confidence: 99%

Two-Term Disjunctions on the Second-Order Cone

Kılınç-Karzan

Yıldız

2014

Integer Programming and Combinatorial Optimization

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Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. In particular, our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kılınç, and Vielma, and by Andersen and Jensen.

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“…For example, a cubic term x i x j x k can be made quadratic with an extra variable and constraint: x i y, where y = x j x k . For more background on QCQP, we refer the reader to [1,2].…”

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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The MILP Road to MIQCP

Cited by 47 publications

References 53 publications

Second-Order-Cone Constraints for Extended Trust-Region Subproblems

Second-Order-Cone Constraints for Extended Trust-Region Subproblems

Two-Term Disjunctions on the Second-Order Cone

Representing quadratically constrained quadratic programs as generalized copositive programs

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