Sufficient matrices and the linear complementarity problem

We pose and answer two questions about solutions of the linear complementarity problem (LCP). The first question is concerned with the conditions on a matrix M which guarantee that for every vector q, the solutions of the LCP (q, M) are identical to the Karush-Kuhn-Tucker points of the natural quadratic program associated with (q, M). In answering this question we introduce the class of "row sufficient" matrices. The transpose of such a matrix is what we call "column sufficient". The latter matrices turn out to furnish the answer to our 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, we discuss the connections of these twonew matrix classes with other well-known matrix classes in linear complementarity theory.