Solution of symmetric linear complementarity problems by iterative methods
 O. L. Mangasarian
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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, GaussSeidel, 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: $$Mx + q > 0 for some x in R^n $$ . Also included are symmetric positivesemidefinite 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.
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 Title
 Solution of symmetric linear complementarity problems by iterative methods
 Journal

Journal of Optimization Theory and Applications
Volume 22, Issue 4 , pp 465485
 Cover Date
 19770801
 DOI
 10.1007/BF01268170
 Print ISSN
 00223239
 Online ISSN
 15732878
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Complementarity problems
 quadratic programming
 mathematical programming
 relaxation methods
 Industry Sectors
 Authors

 O. L. Mangasarian ^{(1)}
 Author Affiliations

 1. Computer Sciences Department, University of Wisconsin, Madison, Wisconsin