Matrix multisplitting methods with applications to linear complementarity problems: Parallel asynchronous methods

We consider parallel matrix multisplitting methods for solving linear complementarity problem that finds a real vector z is an element of R-n such that Mz + q greater than or equal to 0, z greater than or equal to 0 and z(T)(Mz+q) = 0, where M is an element of R-nxn is a given real matrix and q is an element of R-n a given real vector. The recently developed parallel asynchronous multisplitting iterative methods based on fixed-point transformation of the problem, explicit projection of the system and implicit splittings of the matrix are reviewed; their asymptotic convergence properties for some typical matrix class are discussed; and their internal relationships are studied. Therefore, systematic algorithmic models in the sense of multisplitting and reliable theoretical guarantees in the sense of asymptotic convergence are presented for solving the large sparse linear complementarity problems on modem high-speed multiprocessor systems. This paper is a continuity of the recent work of Bai and Evans [18], which includes the parallel synchronous and chaotic matrix multisplitting iterative methods and their convergence theories.