A new subclass of Q0-matrix in linear complementarity theory

In this article, we introduce a new matrix class L¯(d) (a subclass of Q0-matrices which are obtained as a limit of a sequence of L(d)-matrices) such that for any A in this class, a solution to LCP(q,A) exists if LCP(q,A) is feasible, and Lemke's algorithm will find a solution or demonstrate infeasibility. We present a counterexample to show that an L¯(d)-matrix need not be an L(d)-matrix. We also show that if A∈L¯(d), there is an even number of solutions for any nondegenerate vector q. An application of this new matrix class that arises from general quadratic programs and polymatrix games belongs to this class. Finally, we present an example related to the existence of equilibrium in polymatrix games.