The second-order cone eigenvalue complementarity problem. (2nd January 2016)

• Record Type: Journal Article
• Authors: Fernandes, Luís M.; Fukushima, Masao; Júdice, Joaquim J.; Sherali, Hanif D.
• Is Part Of: Optimization methods and software. Volume 31:Number 1(2016)
• Page Start: 24
• Page End: 52
• Publication Date: 2016-01-02
• Subjects: eigenvalue problems -- complementarity problems -- nonlinear programming -- global optimization; 90B60 -- 90C33 -- 90C30 -- 90C26; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2015.1040156
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : The eigenvalue complementarity problem (EiCP) differs from the traditional eigenvalue problem in that the primal and dual variables belong to a closed and convex cone K and its dual, respectively, and satisfy a complementarity condition. In this paper we investigate the solution of the second-order cone EiCP (SOCEiCP) where K is the Lorentz cone. We first show that the SOCEiCP reduces to a special Variational Inequality Problem on a compact set defined by K and 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.