Sufficient Matrices and the Linear Complementarity Problem.

This paper poses and answers two questions about solutions of the linear complementary problem LCP. The first question is concerned with the conditions on a square matrix M which guarantee that for every vector q, the solutions of LCP q, M ar identical to the Karush-Kuhn-Tucker points of the natural quadratic program associated with q, M. In answering this question the authors introduce the class of row sufficient matrices. The transpose of such a matrix is what is called column sufficient. The latter matrices turn out to furnish the answer to the second question which asks for the conditions on M under which the solution set of q, M is convex for every q. In addition to these two main results, this paper discusses the connections of these two new matrix classes with other well-known matrix classes in linear complementarity theory.