The Linear Complementarity Problem

This study centers on the task of efficiently finding a solution of the linear complementarity problem: Ix - My = q, x \ge 0, y \ge 0, x \perp y. The main results are: (1) It is shown that Lemke's algorithm will solve (or show no solution exists) the problem for M \in L where L is a class of matrices, which properly includes (i) certain copositive matrices, (ii) certain matrices with nonnegative principal minors, (iii) matrices for bimatrix games. (2) If M \in L, if the system Ix - My = q, x \ge 0, y \ge 0 is feasible and nondegenerate, then the corresponding linear complementarity problem has an odd number of solutions. If M \in L and q > 0 then the solution is unique. (3) If for some M and every q \ge 0 the problem has a unique solution then M \in L and the problem has a solution for every q. (4) If M has nonnegative principal minors and if the linear complementarity with M and q has a nondegenerate complementary solution then the solution is unique. (5) If y T My + y T q is bounded below on y \ge 0 then the linear complementarity problem with M and q has a solution and Lemke's algorithm can be used to find such a solution. If, in addition, the problem is nondegenerate, then it has an odd number of solutions. (6) A procedure based on Lemke's algorithm is developed which either computes stationary points for general quadratic programs or else shows that the program has no optimum. (7) If a quadratic program has an optimum and satisfies a nondegeneracy condition then there are an odd number of stationary points.