# Solution of symmetric linear complementarity problems by iterative methods

Contributed Papers

DOI: 10.1007/BF01268170

- Cite this article as:
- Mangasarian, O.L. J Optim Theory Appl (1977) 22: 465. doi:10.1007/BF01268170

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## Abstract

A unified treatment is given for iterative algorithms for the solution of the symmetric linear complementarity problem:, where. Also included are symmetric positive-semidefinite matrices satisfying this qualification, symmetric, strictly copositive matrices, and symmetric positive matrices. Furthermore, when

$$Mx + q \geqslant 0, x \geqslant 0, x^T (Mx + q) = 0$$

*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 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:$$Mx + q > 0 for some x in R^n $$

*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.### Key Words

Complementarity problemsquadratic programmingmathematical programmingrelaxation methods## Copyright information

© Plenum Publishing Corporation 1977