Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Abstract

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map fC -> H is a sum of two continuous maps h and g, where h is positively homogeneous of degree gamma (>= 0) on C and g(x)=o(

x

gamma) as

x

->infinity in C. Given such a map f, a nonempty closed convex subset K of C, and a q is an element of H, we consider the variational inequality problem, VI(f,K,q), of finding an x is an element of K such that f(x)+q,x-x >= 0 for all x is an element of K. In this paper, we establish some results connecting the variational inequality problem VI(f,K,q) and the cone complementarity problem CP(f infinity,K infinity,0), where f infinity:=h is the homogeneous part of f and K infinity is the recession cone of K. We show, for example, that VI(f,K,q) has a nonempty compact solution set for every q when zero is the only solution of CP(f infinity,K infinity,0) and the (topological) index of the map x?x-Pi K infinity(x-G(x)) at the origin is nonzero, where G is a continuous extension of f infinity to H. As a consequence, we generalize a complementarity result of Karamardian(J Optim Theory Appl 19:227-232, 1976) formulated for homogeneous maps on proper cones to variational inequalities. The results above extend some similar results proved for affine variational inequalities and for polynomial complementarity problems over the nonnegative orthant in Rn. As an application, we discuss the solvability of nonlinear equations corresponding to weakly homogeneous maps over closed convex cones. In particular, we extend a result of Hillar and Johnson (Proc Am Math Soc 132:945-953, 2004) on the solvability of symmetric word equations to Euclidean Jordan algebras.