On eigenvalues induced by a cone constraint.
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.
scientific article; zbMATH DE number 1993126