Solution of symmetric linear complementarity problems by iterative methods

Abstract

Abstract.A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem:where M is a given n × n symmetric real matrix and q is a given n × 1 vector. A general algorithm is proposed in which relaxation may be performed both before and after projection on the nonnegative orthant. The algorithm includes, as special cases, extensions of the Jacobi, Gauss-Seidel, and nonsymmetric and symmetric successive overrelaxation methods for solving the symmetric linear complementarity problem. It is shown first that any accumulation point of the iterates generated by the general algorithm solves the linear complementarity problem. It is then shown that a class of matrices, for which the existence of an accumulation point that solves the linear complementarity problem is guaranteed, includes symmetric copositive plus matrices which satisfy a qualification of the type:for some x in R n.Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, when M is symmetric, copositive plus, and has nonzero principal subdeterminants, it is shown that the entire sequence of iterates converges to a solution of the linear complementarity problem.

Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities

“…One method for solving (6.1) is based on matrix splitting (see [22,25,31] ). In this method the matrix M is decomposed into the sum of two matrices M = K+L, and, given the current iterate p(t), the next iterate p(t+l) is computed to be a solution of the following linear complementarity problem…”

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

“…In particular, we apply this algorithm to linear complementarity problems (for which X is the non-negative orthant) to obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms [Man77], [Pan84], [LiP87], this algorithm requires no additional assumption (such as symmetry) on the problem data for convergence. We also apply this algorithm to generalized linear/quadratic programming problems to obtain a new decomposition method for solving these problems.…”

“…This problem, called the linear complementarity problem, is a classical problem in optimization (see [BaC78], [CGL80], [Man77], [Pan84]). …”

Book Review: The linear complementarity problem

“…An analogous splitting for the LCP 0 € Bxi+l + Cx' + q + 7V+(x'+1) results in a sequence of problems LCP(2?, Cx' + q) to solve. The seminal work on iterative approaches for LCP is due to Mangasarian [7], although the use of the terminology of splitting was introduced by Pang [9]. The theory relating to these kinds of splittings is developed in Chapter 5 and is still a subject of much current research in the field.…”