Improved modulus-based matrix splitting iteration methods for a class of horizontal implicit complementarity problems

Abstract

Recently, the modulus-based matrix splitting (MMS) iteration method has been successfully extended to solve the horizontal implicit complementarity problem (HICP). However, due to the existence of two system matrices and one nonlinear term, in addition to an equivalent fixed-point equation of the HICP, a system of linear equations has to be solved at each iteration to satisfy both the nonnegative constraint and the orthogonal constraint. This costs very expensive. In this paper, by introducing an additional judgement criterion after obtaining the approximate solution, a class of improved MMS (IMMS) iteration methods are proposed. By suitably choosing the judgement criterion, the IMMS iteration methods do not need to solve such system of linear equations at each iteration and thus greatly improves the computing efficiency. Convergence conditions of the IMMS iteration methods are studied when the system matrices are positive definite matrices and H+-matrices, respectively. Finally, two numerical examples are presented to show the effectiveness of the proposed IMMS iteration methods and their advantages over the existing MMS iteration methods when solving the HICP.