An alternating direction method of multipliers for the eigenvalue complementarity problem. (4th May 2021)

• Record Type: Journal Article
• Authors: Júdice, Joaquim J.; Fukushima, Masao; Iusem, Alfredo; Martinez, J. M.; Sessa, Valentina
• Is Part Of: Optimization methods and software. Volume 36:Number 2/3(2021)
• Page Start: 337
• Page End: 370
• Publication Date: 2021-05-04
• Subjects: Complementarity problems -- eigenvalue problems -- nonlinear programming -- global optimization; 90C30 -- 90C33 -- 90C26 -- 93B60; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1734804
• Languages: English
• ISSNs: 1055-6788

Abstract

ABSTRACT: We introduce an Alternating Direction Method of Multipliers (ADMM) for finding a solution of the nonsymmetric Eigenvalue Complementarity Problem (EiCP). A simpler version of this method is proposed for the symmetric EiCP, that is, for the computation of a Stationary Point (SP) of a Standard Fractional Quadratic Program. The algorithm is also extended for the computation of an SP of a Standard Quadratic Program (StQP). Convergence analyses of these three versions of ADMM are presented. The main computational effort of ADMM is the solution of a Strictly Convex StQP, which can be efficiently solved by a Block Principal Pivoting algorithm. Furthermore, this algorithm provides a stopping criterion for ADMM that improves very much its efficacy to compute an accurate solution of the EiCP. Numerical results indicate that ADMM is in general very efficient for solving symmetric EiCPs in terms of the number of iterations and computational effort, but is less efficient for the solution of nonsymmetric EiCPs. However, ADMM is able to provide a good initial point for a fast second-order method, such as the so-called Semismooth Newton method. The resulting hybrid ADMM and SN algorithm seems to be quite efficient in practice for the solution of nonsymmetric EiCPs.