The second-order cone eigenvalue complementarity problem.

The eigenvalue complementarity problem (EiCP) differs from the traditional eigenvalue problem in that the primal and dual variables belong to a closed and convex coneKand its dual, respectively, and satisfy a complementarity condition. In this paper we investigate the solution of the second-order cone EiCP (SOCEiCP) whereKis the Lorentz cone. We first show that the SOCEiCP reduces to a special Variational Inequality Problem on a compact set defined byKand a normalization constraint. This guarantees that SOCEiCP has at least one solution, and a new enumerative algorithm is introduced for finding a solution to this problem. The method is based on finding a global minimum of an appropriate nonlinear programming (NLP) formulation of the SOCEiCP using a special branching scheme along with a local nonlinear optimizer that computes stationary points on subsets of the feasible region of NLP associated with the nodes generated by the algorithm. A semi-smooth Newton's method is combined with this enumerative algorithm to enhance its numerical performance. Our computational experience illustrates the efficacy of the proposed techniques in practice. [ABSTRACT FROM AUTHOR]