Copositive Optimization

Bomze, Immanuel M.

doi:10.1007/978-0-387-74759-0_99

Cited by 13 publications

(19 citation statements)

References 20 publications

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“…So we need to approximate them by so-called hierarchies, i.e., a sequence of tractable cones K d such that K d ⊂ K d+1 ⊂ C , where d is the level of the hierarchy and ∞ d=0 K d = [C ] • , i.e., every strictly copositive matrix is contained in K d for some d. On the dual side, K d are also tractable, K d+1 ⊂ K d , and ∞ d=0 K d = C contains no matrix which is not completely positive. For brevity of exposition, assume that z * CD = z * CP and further assume that strong duality also holds for the following approximation: [11]. Many of these involve linear or psd constraints of matrices of order n d+2 , e.g., the seminal ones proposed in [34,41].…”

Section: Possible Algorithmic Implicationsmentioning

confidence: 99%

“…So quite naturally we are led to our first SDP in (a) or copositive optimization problem in (b): optimize a linear function of a variable μ under the constraint that a matrix affine-linear in μ is either psd or copositive. More generally, in a copositive optimization problem (for surveys, see, e.g., [8,11,18,24]), we are given r ∈ R m as well as m + 2 symmetric matrices {M 0 , . .…”

Section: Consequences Of An Elementary Observationmentioning

confidence: 99%

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Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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We study nonconvex quadratic minimization problems under (possibly nonconvex) quadratic and linear constraints, characterizing both Lagrangian and semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be novel), we show that the semi-Lagrangian dual is equivalent to a natural copositive relaxation (and this has apparently not been observed before). This way, we arrive at conic bounds tighter than the usual Lagrangian dual (and thus than the SDP) bounds. Any of the known tractable inner approximations of the copositive cone can be used for this tightening, but in particular, the above-mentioned characterization with explicit linear constraints is a natural, much cheaper, relaxation than the usual zero-order approximation by doubly nonnegative (DNN) matrices and still improves upon the Lagrangian dual bounds. These approximations are based on LMIs on matrices of basically the original order plus additional linear constraints (in contrast to more familiar sum-of-squares or moment approximation hierarchies) and thus may have merits in particular for large instances where it is important to employ only a few inequality constraints (e.g., n instead of n(n−1) 2 for the DNN relaxation). Further, we specify sufficient conditions for tightness of the semi-Lagrangian relaxation and show that copositivity of the slack matrix guarantees global optimality for KKT points of this problem, thus significantly improving upon a well-known secondorder global optimality condition.

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Section: Possible Algorithmic Implicationsmentioning

confidence: 99%

Section: Consequences Of An Elementary Observationmentioning

confidence: 99%

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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“…A recent survey on copositive optimization is offered by [57], while [77] and [74] provide reviews on copositivity with less emphasis on optimization. Bomze [16] and Busygin [37] provided entries in the most recent edition of the Encyclopedia of Optimization. Recent book chapters with some character of a survey on copositivity from an optimization viewpoint are [17,Section 1.4] and [34].…”

Section: Surveys Reviews Entries Book Chaptersmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

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135

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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“…Recent surveys on copositive optimization are offered by [108] and [43], while [152] and [141] provide reviews on copositivity with less emphasis on optimization. [41] and [71]…”

Section: Surveys Reviews Entries Book Chaptersmentioning

confidence: 99%

Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

2011

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Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone C of all completely positive symmetric n×n matrices (which can be factorized into F F ⊤ , where F is a rectangular matrix with no negative entry), and its dual cone C * , which coincides with the cone of all copositive matrices (those which generate a quadratic form taking no negative value over the positive orthant). We provide structural algebraic properties of these cones, and numerous (counter-)examples which demonstrate that many relations familiar from semidefinite optimization may fail in the copositive context, illustrating the transition from polynomial-time to NP-hard worst-case behaviour. In course of this development we also present a systematic construction principle for non-attainability phenomena, which apparently has not been noted before in an explicit way. Last but not least, also seemingly for the first time, a somehow systematic clustering of the vast and scattered literature is attempted in this paper.

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Copositive Optimization

Cited by 13 publications

References 20 publications

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Copositive optimization – Recent developments and applications

Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

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