Almost-Complementary Paths in the Generalized Complementarity Problem

Abstract

If K is a closed convex cone in Rn, K+ is its polar cone and w is a continuous mapping from K into Rn, then the generalized complementarity problem is to find x such that x ∈ K, w(x) ∈ K+, and the inner product xw(x) = 0. Using a fixed-point algorithm it is shown that there is a connected set of almost-complementary points which joins the origin to either a complementary point or to infinity. This is a generalization of the almost-complementary path generated by Lemke for the linear complementarity problem (K = Rn+ and w affine).