A multisplitting method for symmetric linear complementarity problems

Over the years, many methods for solving the linear complementarity problem (LCP) have been developed. Most of these methods have their origin in solving a system of linear equations. In particular, much attention has recently been paid on the class of iterative methods called the splitting method, which is an extension of the matrix splitting method for solving a system of linear equations such as Jacobi, Gauss-Seidel and SOR methods. Furthermore, as a method for solving a system of linear equations, O'Leary and White have proposed a parallel iterative method called the multisplitting method. This method makes use of a set of different splittings of the coefficient matrix, which may be dealt with independently of each other. The results obtained from those splitting iterations are combined to define the multisplitting iterates. Thus, the method may be effectively implemented on multiprocessors. In this paper, we extend the idea of the multisplitting to the symmetric LCP. In particular, we establish some convergence results for the multisplitting method, which generalize the corresponding convergence results for the splitting method for LCP. We also report some computational results with the proposed method.