Cone-constrained eigenvalue problems: theory and algorithms

Costa, A. Pinto da; Seeger, Alberto

doi:10.1007/s10589-008-9167-8

Cited by 85 publications

(55 citation statements)

References 12 publications

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“…in [61,62,70,95,101]. An interesting application to Simpson's paradox can be found in [69], while the connection of copositivity with conic geometry and angles is discussed, mostly from an optimization perspective, in [5,65,78,110].…”

Section: Applications Of Copositive Optimizationmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

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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Section: Applications Of Copositive Optimizationmentioning

confidence: 99%

Copositive optimization – Recent developments and applications

Bomze

2012

European Journal of Operational Research

136

117

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“…The Jacobian matrix J EB (z) has the same structure as (23), except that E and F are slightly different: I r must be changed by a diagonal matrix with nonzero entries…”

Section: Propositionmentioning

confidence: 99%

“…We focus the attention to the more complicated case in which A is asymmetric. The Scaling-and-Projection Algorithm and the Power Iteration Method have been introduced and studied in [22,23]. Global optimization and Branch-and-Bound techniques have been explored by Júdice et al [14].…”

mentioning

confidence: 99%

A nonsmooth algorithm for cone-constrained eigenvalue problems

Adly

Seeger

2009

Comput Optim Appl

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International audienceWe study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form Kx(Ax−Bx)K+ Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. The symbol K refers to a closed convex cone in the Euclidean space ℝ n and (A,B) is a pair of possibly asymmetric matrices of order n. Special attention is paid to the case in which K is the nonnegative orthant of ℝ n . The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail

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“…For the general theory behind Problems 1 and 2, the reader is conveyed to the references [7,11,15,16]. This note focuses only on numerical issues.…”

Section: Problemmentioning

confidence: 99%

“…The case of a variable scaling factor will be commented in Section 4. The next convergence result is taken from [11]. (b) {y t } t≥0 converges toȳ = Ax −λBx.…”

Section: Problemmentioning

confidence: 99%

Numerical resolution of cone-constrained eigenvalue problems

Costa¹,

Seeger²

2009

Comput.appl.math.

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Abstract. Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity systemThis problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.Mathematical subject classification: 65F15, 90C33.

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Cone-constrained eigenvalue problems: theory and algorithms

Abstract: Complementarity conditions, Generalized eigenvalue problems, Convex cones,

Cited by 85 publications

References 12 publications

Copositive optimization – Recent developments and applications

Copositive optimization – Recent developments and applications

A nonsmooth algorithm for cone-constrained eigenvalue problems

Numerical resolution of cone-constrained eigenvalue problems

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