A Variational Approach to Copositive Matrices

Hiriart-Urruty, Jean-Baptiste; Seeger, Alberto

doi:10.1137/090750391

Cited by 123 publications

(81 citation statements)

References 85 publications

(107 reference statements)

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“…if A is symmetric, then When m = 2, both λ(A) and µ(A) are the smallest Pareto eigenvalue of a matrix A, denote by λ(A). For more details on Pareto eigenvalue of a matrix, see Seeger [40], Seeger, Torki [41] and Hiriart-Urruty, Seeger [42]. Then the following conclusions are easy to obtain.…”

Section: Boundedness Of Solution Set Of Tcp(a Q)mentioning

confidence: 88%

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Song

2016

Optim Lett

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In this paper, we present the boundedness of solution set of tensor complementarity problem defined by a strictly semi-positive tensor. For strictly semi-positive tensor, we prove that all H + (Z + )-eigenvalues of each principal sub-tensor are positive. We define two new constants associated with H + (Z + )-eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too.

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Section: Boundedness Of Solution Set Of Tcp(a Q)mentioning

confidence: 88%

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Song

2016

Optim Lett

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“…While the time may not yet be ripe for writing up the final standard text book in this domain, several authors nonetheless bravely took the challenge of providing an overview, thereby aiming at a rapidly moving target. 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.…”

Section: Surveys Reviews Entries Book Chaptersmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

132

113

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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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“…It can be checked by means of the so-called Pareto eigenvalues [33], but computing those is not doable in polynomial time. Spectral properties of copositive matrices provide some information and are discussed in [38].…”

Section: Copositivity Criteria Based On Structural Matrix Propertiesmentioning

confidence: 99%

“…Spectral properties of copositive matrices provide some information and are discussed in [38]. For dimensions up to four, explicit descriptions are available [33]. For example, a symmetric 2 × 2 matrix A is copositive if and only if its entries fulfill a 11 ≥ 0, a 22 ≥ 0 and a 12 + √ a 11 a 22 ≥ 0, see [1].…”

Section: Copositivity Criteria Based On Structural Matrix Propertiesmentioning

confidence: 99%

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Copositive Programming – a Survey

Dür

2010

Recent Advances in Optimization and Its Applications in Engineering

167

176

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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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A Variational Approach to Copositive Matrices

Cited by 123 publications

References 85 publications

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

Copositive optimization – Recent developments and applications

Copositive Programming – a Survey

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