On eigenvalues induced by a cone constraint.
type: Item
instance of: scholarly article
title: On eigenvalues induced by a cone constraint.
zbMATH Open document ID: 1046.15008
DOI: 10.1016/S0024-3795(03)00553-6
author: Mounir Torki
author: Alberto Seeger
published in: Linear Algebra and its Applications
publication date: 2003-10-15T00:00:00Z
review text: The authors analyze the spectral properties of a given \(n\times n\) real matrix \(A\) depending heavily on the cone \(K\subset\mathbb{R}^n\), that is used as a constraint set. More precisely, the authors focuss on the problem of analyzing the main properties of the mapping \(\sigma(A,K)\) that denotes the spectrum of \(A\) relative to \(K\). That is, \[ \sigma(A,K):= \{\lambda\in \mathbb{R}:(\lambda, x)\text{ solves (P) for some }x\}, \] where (P) is the problem consisting of find a number \(\lambda\in \mathbb{R}\) and a nonzero vector \(x\in \mathbb{R}^n\) such that \(x\in K\), \(Ax-\lambda x\in K^+\), and \(\langle x,Ax-\lambda x\rangle= 0\). For the purpose, they use some concept concerned to \textit{G. Beer} [Topologies on closed and closed convex sets (Mathematics and its Applications 268, Kluwer Academic Publishers, Dordrecht) (1993; Zbl 0792.54008)]. On the other side, they also discuss some structured differences existing between the polyhedral case (that is, when \(K\) is finitely generated), and the nonpolyhedral case. For the polyhedral case, the authors use some important aspect provided by \textit{A. Seeger} [Linear Algebra Appl. 292, 1--14 (1999; Zbl 1016.90067)]. For the case that \(K\) is nonpolyhedral, the authors make an special mention to the Lorentz cone.
reviewed by: J. Rafael Sendra
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1993126
zbMATH Keywords: eigenvalue
zbMATH Keywords: linear complementarity problem
zbMATH Keywords: polyhedral cone
zbMATH Keywords: Lorentz cone
MaRDI profile type: Publication
cites work: Theory of cones
cites work: Q4277838
cites work: Q3943082
cites work: Q4747150
cites work: Q4321748
cites work: Q4050739
cites work: Q5547252
cites work: Q4194725
cites work: Variational methods for nonlinear eigenvalue inequalities
cites work: A new method for obtaining eigenvalues of variational inequalities based on bifurcation theory
cites work: A new method for obtaining eigenvalues of variational inequalities: operators with multiple eigenvalues
cites work: Q4551020
cites work: Positive operators on the \(n\)-dimensional ice cream cone
cites work: Eigenvalue problems for nonlinear elliptic variational inequalities on a cone
cites work: Variational Analysis
cites work: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions
cites work: Exponential Nonnegativity on the Ice Cream Cone
cites work: Continuity of Some Convex-Cone-Valued Mappings
cites work: Lectures on Polytopes
description: scientific article; zbMATH DE number 1993126