On global optimization with indefinite quadratics

Fampa, Marcia; Lee, Jon; Melo, Wendel

doi:10.1007/s13675-016-0079-6

Cited by 12 publications

(24 citation statements)

References 26 publications

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“…Constraints (12) and 13are known as the McCormick envelopes and provide a relaxation of the product of two continuous variables. The product of binary and continuous variables is exactly linearized by constraints (14)- (16).…”

Section: Normalized Multiparametric Disaggregationmentioning

confidence: 99%

“…Definition 1 For every p, EQUIV p is defined as the problem of minimizing the objective function (17), subject to the constraints (5)-(7), (10), (11), (14)- (16), (18), and (19).…”

Section: Normalized Multiparametric Disaggregationmentioning

confidence: 99%

“…This method is known as the difference of convex functions (DC) approach. Fampa et al [16] proposed several approaches for decomposing the quadratic functions as sums of a convex and a concave function. Another possible strategy is to determine envelopes for the quadratic terms over regions other than rectangles [27].…”

Section: Introductionmentioning

confidence: 99%

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Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

et al. 2018

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We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.

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Section: Normalized Multiparametric Disaggregationmentioning

confidence: 99%

“…Definition 1 For every p, EQUIV p is defined as the problem of minimizing the objective function (17), subject to the constraints (5)-(7), (10), (11), (14)- (16), (18), and (19).…”

Section: Normalized Multiparametric Disaggregationmentioning

confidence: 99%

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Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

et al. 2018

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“…While stationary points of such problems can be obtained by nonlinear programming solvers, global solutions require branching-based schemes. In Section 4.3, inspired by recent research by Fampa et al [16], we present a technique for obtaining global solutions of a nonconvex quadratically constrained quadratic program. Proof.…”

Section: General Uncertain Non-monotone Lcps In This Section We Conmentioning

confidence: 99%

On Robust Solutions to Uncertain Linear Complementarity Problems and their Variants

Xie

Shanbhag

2016

SIAM J. Optim.

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Abstract. A popular approach for addressing uncertainty in variational inequality problems is by solving the expected residual minimization (ERM) problem [9,10]. This avenue necessitates distributional information associated with the uncertainty and requires minimizing nonconvex expectation-valued functions. We consider a distinctly different approach in the context of uncertain linear complementarity problems with a view towards obtaining robust solutions. Specifically, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we show that under specified assumptions on the uncertainty sets, the robust solutions to uncertain monotone linear complementarity problem can be tractably obtained through the solution of a single convex program. We also define uncertainty sets that ensure that robust solutions to non-monotone generalizations can also be obtained by solving convex programs. More generally, robust counterparts of uncertain non-monotone LCPs are proven to be low-dimensional nonconvex quadratically constrained quadratic programs. We show that these problems may be globally resolved by customizing an existing branching scheme. We further extend the tractability results to include uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity and traffic equilibrium problems suggest that the presented avenues hold promise.

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“…DC decompositions have been extensively used in the literature to generate convex quadratic relaxations of nonconvex quadratic problems. See, for example, [10] and references therein. Unlike the approach used in DC decompositions, we do not necessarily decompose x T Qx as a difference of convex functions, or equivalently, as a sum of a convex and a concave function.…”

Section: Introductionmentioning

confidence: 99%