Solving Symmetric and Positive Definite Second-Order Cone Linear Complementarity Problem by A Rational Krylov Subspace Method

TL;DR

This paper introduces a new rational Krylov subspace method to efficiently solve symmetric positive definite second-order cone linear complementarity problems, improving accuracy and robustness over existing approaches.

Contribution

The paper presents a novel two-stage rational Krylov subspace algorithm specifically designed for symmetric positive definite SOCLCPs, combining extended and multiple-pole projections.

Findings

Method demonstrates high efficiency in numerical tests.

Algorithm shows robustness across various problem types.

Achieves accurate solutions with fewer iterations.

Abstract

The second-order cone linear complementarity problem (SOCLCP) is a generalization of the classical linear complementarity problem. It has been known that SOCLCP, with the globally uniquely solvable property, is essentially equivalent to a zero-finding problem in which the associated function bears much similarity to the transfer function in model reduction [{\em SIAM J. Sci. Comput.}, 37 (2015), pp.~A2046--A2075]. In this paper, we propose a new rational Krylov subspace method to solve the zero-finding problem for the symmetric and positive definite SOCLCP. The algorithm consists of two stages: first, it relies on an extended Krylov subspace to obtain rough approximations of the zero root, and then applies multiple-pole rational Krylov subspace projections iteratively to acquire an accurate solution. Numerical evaluations on various types of SOCLCP examples demonstrate its efficiency and robustness.