Numerical resolution of cone-constrained eigenvalue problems

Costa, A. Pinto da; Seeger, Alberto

doi:10.1590/s0101-82052009000100003

Cited by 26 publications

(30 citation statements)

References 12 publications

(21 reference statements)

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“…This does not mean that the SMN is systematically better than the methods suggested in [22,23]. As said before, implementing the SNM could be a headache if the cone K is not polyhedral.…”

Section: By Way Of Conclusionmentioning

confidence: 93%

“…Hence, we go beyond the framework of Sect. 2 and of reference [22]. Recall that the lineality of a convex cone K in R n is defined as the integer…”

Section: Coping With Unpointednessmentioning

confidence: 99%

“…What is safe to say, at least, is this: -While working with some particular examples, like the one of Sect. 2.2.1, the SNM was able to find the solutions that were left undetected by the Scaling-andProjection Algorithm [23] and by the Power Iteration Method [22,23].…”

Section: By Way Of Conclusionmentioning

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].…”

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confidence: 99%

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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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“…This does not mean that the SMN is systematically better than the methods suggested in [22,23]. As said before, implementing the SNM could be a headache if the cone K is not polyhedral.…”

Section: By Way Of Conclusionmentioning

confidence: 93%

“…Hence, we go beyond the framework of Sect. 2 and of reference [22]. Recall that the lineality of a convex cone K in R n is defined as the integer…”

Section: Coping With Unpointednessmentioning

confidence: 99%

Section: By Way Of Conclusionmentioning

confidence: 99%

mentioning

confidence: 99%

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A nonsmooth algorithm for cone-constrained eigenvalue problems

Adly

Seeger

2009

Comput Optim Appl

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“…All authors read and approved the final manuscript. 1 Department of Mathematics, Tianjin University, Tianjin, 300072, China.…”

Section: Competing Interestsmentioning

confidence: 99%

An alternative approach for a distance inequality associated with the second-order cone and the circular cone

2016

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It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.

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Power iteration and inverse power iteration for eigenvalue complementarity problem

Abdi

Shakeri

2019

Numerical Linear Algebra App

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Summary In this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.

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Numerical resolution of cone-constrained eigenvalue problems

Cited by 26 publications

References 12 publications

A nonsmooth algorithm for cone-constrained eigenvalue problems

A nonsmooth algorithm for cone-constrained eigenvalue problems

An alternative approach for a distance inequality associated with the second-order cone and the circular cone

Power iteration and inverse power iteration for eigenvalue complementarity problem

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