A note on generalised linear complementarity problems

Abstract

Given ann×nmatrixA, ann-dimensional vectorq, and a closed, convex coneSofRn, the generalized linear complementarity problem considered here is the following: find az∈Rnsuch thatwheres* is the polar cone ofS. The existence of a solution to this problem for arbitrary vectorqhas been established both analytically and constructively for several classes of matricesA. In this note, a new class of matrices, denoted byJ, is introduced.Ais aJ-matrix ifThe new class can be seen to be broader than previously studied classes. We analytically show that for anyAin this class, a solution to the above problem exists for arbitrary vectorq. This is achieved by using a result on variational inequalities.