A Characterization of the Constant Parity Property of the Number of Solutions to the Linear Complementarity Problem

We consider the linear complementarily problem: Given an $m \times m$ matrix M and a real m-vector q, find real m-vectors x and y which solve (i) $x = My + q,x\geqq 0,y\geqq 0$, (ii) $x^T y = 0$. In this paper we prove the necessary and sufficient conditions for the constant parity of the number of solutions to (i)–(ii) for all q for which every solution is nondegenerate. As a consequence we also derive the results of Murty [10], Karamardian [7] and Saigal [11].