A Partial Characterization of a Class of Matrices Defined by Solutions to the Linear Complementarity Problem

In this paper we analyze the class of matrices, Q₀, characterized as all n × n matrices, M, for which the linear complementarity problem, w = Mz + q, w, z ≥ 0, w′z = 0, has a solution whenever q is feasible. The method is to analyze the faces of the linear cone. Cone[I, −M] where I is the n × n identity. Essentially, a labelling requirement for the faces is shown necessary for M to be in Q₀. With proper restrictions and nondegenerate assumptions sufficiency is also shown. These results arc related to the signs of the minors of M and some conjectures in this area are also discussed.