Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f:C-->H is a sum of two continuous maps h and g, where h is positively homogeneous of (positive) degree gamma on C and g(x)/||x||^gamma-->0 as ||x||-->infinity in C. Given such a map f, a nonempty closed convex subset K of C, and a q in H, we consider the variational inequality problem VI(f,K,q). In this paper, using degree theory, we establish some results connecting the variational inequality problem VI(f,K,q) and the cone complementarity problem CP(h,Kinfty,0), where Kinfty is the recession cone of K. As a consequence, we generalize a complementarity result of Karamardian 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 R^n. 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 on the solvability of symmetric word equations to Euclidean Jordan algebras.


Mathematical Programming, Series A; https://doi.org/10.1007/s10107-018-1263-7