Extension of the Generalized Complementarity Problem
Romesh Saigal · Mathematics of Operations Research · 1976
Given a closed convex cone K in the n-dimensional Euclidean space ℝ n , its polar cone K + and a point-to-set mapping f from K to subsets of ℝ n , 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 T y = 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.