, Volume 22, Issue 4, pp 465-485

Solution of symmetric linear complementarity problems by iterative methods

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Abstract

A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem: $$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$ , whereM is a givenn×n symmetric real matrix andq is a givenn×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 over-relaxation 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: $$Mx + q > 0 for some x in R^n $$ . Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, whenM 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.

This research was supported by the Science Research Council, Grant No. B/RG/4079.7, by the National Science Foundation, Grant No. DCR-74-20584, and by the Wisconsin Alumni Research Foundation.
The author is indebted to C. Elliott for stimulating discussions on the subject of this paper and to R. R. Meyer for suggestions for improving the paper and, in particular, for suggesting the use of an arbitrary constantc, instead of zero, in Lemma 2.3.
This paper was written while the author was a Senior Research Fellow, Oxford University Computing Laboratory, Oxford, England.