Copositivity and constrained fractional quadratic problems

Amaral, Paula; Bomze, Immanuel M.; Júdice, Joaquim J.

doi:10.1007/s10107-013-0690-8

Cited by 28 publications

(26 citation statements)

References 48 publications

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“…Taking into account that 3 , it follows that also {s k } is a bounded sequence. By passing to subsequences, we may assume that…”

Section: Proposition 42 Suppose That There Existmentioning

confidence: 99%

“…Copositive optimization is a special case of convex conic optimization (namely, to minimize a linear function over a cone subject to linear constraints). By now, equivalent copositive reformulations for many important problems are known, among them (non-convex, mixed-binary, fractional) quadratic optimization problems under a mild assumption [2,3,13], and some special optimization problems under uncertainty [4,18,32,37]. In particular, it has been shown in [7] that, for quadratic optimization problems with additional nonnegative constraints, copositive relaxations (and its tractable approximations) provides a tighter bound than the usual Lagrangian relaxation.…”

Section: Introductionmentioning

confidence: 99%

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Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

Bomze

Jeyakumar

2018

J Glob Optim

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We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush-KuhnTucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are exact for an extended CDT (two-ball trust-region) problem. Importantly, the sufficient conditions can be verified by solving linear optimization problems.

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“…Taking into account that 3 , it follows that also {s k } is a bounded sequence. By passing to subsequences, we may assume that…”

Section: Proposition 42 Suppose That There Existmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

Bomze

Jeyakumar

2018

J Glob Optim

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“…This NP-hard problem also arises when studying the repair of inconsistent linear systems, for details see [1]. Now define the symmetric (n + 1) × (n + 1) matrices…”

Section: The Fractional Quadratic Casementioning

confidence: 99%

“…Then [113] (for the special case x ∈ R n + : Ax = a = ∆) and [1] showed under (6) that (5) can be written as the completely positive problem:…”

Section: The Fractional Quadratic Casementioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

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136

117

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Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.

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“…Many other formulations of nonconvex programs and applications of the MPLCC have been discussed in the past several years [1,2,10,11,19,22,33,41,40,55,60,61,63,67,70,73,76,80,87]. In this paper we surveyed the most important applications of the MPLCC and formulations of problems as MPLCCs with special emphasis on bilevel, bilinear and nonconvex quadratic programming problems and the eigenvalue complementarity problem.…”

Section: The Mplcc Is Called a Linear (Quadratic) Programming Problemmentioning

confidence: 99%

Optimization With Linear Complementarity Constraints

Júdice

2014

Pesqui. Oper.

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ABSTRACT. A Mathematical Program with Linear Complementarity Constraints (MPLCC)is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications in several areas of science, engineering and economics and is also an important tool for the solution of some NP-hard structured and nonconvex optimization problems, such as bilevel, bilinear and nonconvex quadratic programs and the eigenvalue complementarity problem. In this paper some of the most relevant applications of the MPLCC and formulations of nonconvex optimization problems as MPLCCs are first presented.Algorithms for computing a feasible solution, a stationary point and a global minimum for the MPLCC are next discussed. The most important nonlinear programming methods, complementarity algorithms, enumerative techniques and 0 − 1 integer programming approaches for the MPLCC are reviewed. Some comments about the computational performance of these algorithms and a few topics for future research are also included in this survey.

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Copositivity and constrained fractional quadratic problems

Cited by 28 publications

References 48 publications

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations

Copositive optimization – Recent developments and applications

Optimization With Linear Complementarity Constraints

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