Modified SSOR Modelling for Linear Complementarity Problems

Where, T z denotes the transpose of the vector z . Many problems in operations research, management science, various scientific computing and engineering areas can lead to the solution of an LCP of the form (1.1). For more details (see, [1,2,3] and the references therein). The early motivation for studying the LCP (M, q) was because the KKT optimality conditions for linear and quadratic programs constitute an LCP of the form (1.1). However, the study of LCP really came into prominence since the 1960s and in several decades, many methods for solving the LCP (M, q) have been introduced. Most of these methods originate from those for the system of the linear equations where these methods may be classified into two principal kinds, direct and iterative methods (see, [1,2,3]). Recently, various authors have suggested different model in the frame of the iterative methods for the above mentioned problem. For example, Yuan and Song in [4], based on the models in [5], proposed a class of modified accelerated over-relaxation (MAOR) methods to solve ( , ) LCP M q . Furthermore, when the system matrix M is an H-matrix they proposed some sufficient conditions for convergence of the MAOR and modified successive over-relaxation (MSOR) methods. Under certain conditions, Li and Dai [6], Saberi Najafi and Edalatpanah [7] and Dehghan and Hajarian [8] also studied generalized accelerated over-relaxation (GAOR) and symmetric successive over-relaxation (SSOR) methods for solving ( , ) LCP M q , respectively. The case that M is Non-Hermitian, Saberi Najafi and Edalatpanah [8,9], proposed some new iterative methods for solving this class of LCP (to see that other iterative methods for ( , ) LCP M q see [11-17] and the references therein). In this paper, we will propose a modification of SSOR method for ( , ) LCP M q . To accomplish of this purpose, SSOR method is coupled with the preconditioning strategy. We also show that our method for solving LCP is superior to the basic SSOR method. Numerical experiments show that the new method is feasible and efficient for solving large sparse linear complementarity problems.