On the number of solutions to the complementarity problem and spanning properties of complementary cones

Abstract The relationship between the number of solutions to the complementarity problem, w = Mz + q , w⩾0, z⩾0, w T z=0 , the right-hand constant vector q and the matrix M are explored. The main results proved in this work are summarized below. The number of solutions to the complementarity problem is finite for all q ϵ R n if and only if all the principal subdeterminants of M are nonzero. The necessary and sufficient condition for this solution to be unique for each q ϵ R n is that all principal subdeterminants of M are strictly positive. When M ⩾0, there is at least one complementary feasible solution for each q ϵ R n if and only if all the diagonal elements of M are strictly positive; and, in this case, the number of these solutions is an odd number whenever q is nondegenerate. If all principal subdeterminants of M are nonzero, then the number of complementary feasible solutions has the same parity (odd or even) for all q ϵ R n which are nondegenerate. Also, if the number of complementary feasible solutions is a constant for each q ϵ R n , then that constant is equal to one and M is a P -matrix. In the cartesian system of coordinates for R n , an orthant is a convex cone generated by a set of n -column vectors in R n , {A. 1 ,…,A. n }, where for each j = 1 to n , A . j is either the j th column vector of the unit matrix of order n (denoted by I. j ) or its negative - I. j . There are thus 2 n orthants in R n , and they partition the whole space. It is interesting to know what properties these orthants possess if we obtain them after replacing - I. j by some given column vector - M. j for j = 1 to n . Orthants obtained in this manner are called complementary cones , and their spanning properties are studied.