Characterizations of bounded solutions of linear complementarity problems

Abstract

A number of equivalent characterizations for the existence and boundedness of solutions of the linear complementarity problem: Mx+q≥0, x≥0, x T(Mx+q)=0 where M is an n×n real matrix and q is an n-vector, are given for the case when M is copositive plus. The special case when M is skew-symmetric covers the linear programming case. One useful characterization of existence and boundedness of solutions is given by solving a simple linear program. Other important characterizations are the Slater constraint qualification and the stability condition that for all arbitrary but sufficiently small perturbations of the data M and q which maintain copositivity plus, the perturbed linear complementarity problem is solvable and its solutions are uniformly bounded. An interesting sufficient condition for boundedness of solutions is that the linear complementarity problem have a nondegenerate vertex solution. Another result is that the subclass ™ of copositive plus matrices for which the linear complementarity problem has a solution for each q in R n, that is ™⊂Q, coincides with the subclass of copositive plus matrices for which the linear complementarity problem has a nonempty bounded solution set for each q in R n.Key wordsLinear Complementarity ProblemSolution SetBoundednessCopositive-plus Matrices