A nonsmooth algorithm for cone-constrained eigenvalue problems

We study several variants of a nonsmooth Newton-type algorithm for solving an eigenvalue problem of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\ni x\perp(Ax-\lambda Bx)\in K^{+}.$$\end{document} 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 ℝn 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 ℝn. The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.