Copositive programming motivated bounds on the stability and the chromatic numbers

Dukanovic, Igor; Rendl, Franz

doi:10.1007/s10107-008-0233-x

Cited by 34 publications

(37 citation statements)

References 28 publications

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“…Similar copositive optimization approaches, among many others, were employed to obtain bounds on the (fractional) chromatic number of a graph [56]. Further applications of copositive optimization to graph theory and other discrete problems can be found, e.g., in [87,121].…”

Section: Combinatorial Problems From a Copositive Perspectivementioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

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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Section: Combinatorial Problems From a Copositive Perspectivementioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

135

117

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“…Dukanovic and Rendl [26] introduce a related copositivity-inspired strengthening of the Lovász ϑ number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number.…”

Section: Combinatorial Problemsmentioning

confidence: 99%

“…A completely positive formulation of the related problem of computing the fractional chromatic number can be found in [26].…”

Section: Combinatorial Problemsmentioning

confidence: 99%

Copositive Programming – a Survey

Dür

2010

Recent Advances in Optimization and Its Applications in Engineering

179

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Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.

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“…One reason is that linear programs over these cones, usually referred as copositive programs, can be used to formulate many hard optimization problems exactly. Examples include (nonconvex) quadratic programs over the standard simplex [BDdK + 00, BK02], the stability number, the (fractional) chromatic number of a graph [dKP02,PVZ07,DR08,GL08], and the quadratic assignment problem [PR09]. Burer [Bur09] proved that a large class of deterministic (nonconvex) quadratic programming problems, possibly including binary variables and complementary constraints, have exact copositive representations.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone

Dong

2013

SIAM J. Optim.

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In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As an application, we consider a class of bounds on the stability number of a graph obtained from the polyhedral approximation hierarchy, and we construct a primal optimal solution with its tensor lifting for each of such linear programs.

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Copositive programming motivated bounds on the stability and the chromatic numbers

Cited by 34 publications

References 28 publications

Copositive optimization – Recent developments and applications

Copositive optimization – Recent developments and applications

Copositive Programming – a Survey

Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone

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