Extension of the Generalized Complementarity Problem

Given a closed convex cone K in the n-dimensional Euclidean space ℝⁿ, its polar cone K⁺ and a point-to-set mapping f from K to subsets of ℝⁿ, the extension of the generalized complementarity problem we consider is to find a vector x in K such that there is a y in f(x) ∩ K⁺ with x^Ty = 0. We prove, using directly a fixed point theorem, with an appropriate condition on f, that the above problem has a solution. We also show that this condition is satisfied by a certain class of mappings f which are encountered in various applications. We further establish “the basic theorem of complementarity” for this problem, which thus extends the previous existence results to the case where K is a pointed cone.