Cone-constrained eigenvalue problems: structure of cone spectra. (English) Zbl 1471.15029

Seeger, Alberto

Set-Valued Var. Anal. 29, No. 3, 605-619 (2021).

Summary: There is a rich literature devoted to the eigenvalue analysis of variational inequalities. Of special interest is the case in which the constraint set of the variational inequality is a closed convex cone. The set of eigenvalues of a matrix \(A\) relative to a closed convex cone \(K\) is called the \(K\)-spectrum of \(A\). Cardinality and topological results for cone spectra depend on the kind of matrices and cones that are used as ingredients. It is important to distinguish for instance between symmetric and nonsymmetric matrices and, on the other hand, between polyhedral and nonpolyhedral cones. However, more subtle subdivisions are necessary for having a better understanding of the structure of cone spectra. This work elaborates on this issue.

MSC:

15B48	Positive matrices and their generalizations; cones of matrices
15A18	Eigenvalues, singular values, and eigenvectors
47A75	Eigenvalue problems for linear operators
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Keywords:

complementarity eigenproblem; cone spectrum; critical values; quadratic forms on cones; nonpolyhedral cones

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