On the Continuity of the Solution Map in Linear Complementarity Problems

The continuity properties of the solution map S(M,q) ↦ S(M,q) are investigated, where S(M,q) denotes the solution set corresponding to the linear complementarily problem LCP $(M,q)$. A Robinson-type upper semicontinuity result is established for S, and a generalization of the Mangasarian-Shiau result concerning the Lipschitzian property of S in the q-variable is proved. It is also shown that when the matrix is positive semidefinite (or more generally a G-matrix), the solution map is Lipschitz continuous with respect to the q-vector if and only if the matrix is a P-matrix.