Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

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Abstract

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map \(f\,{:}\,C\rightarrow H\) is a sum of two continuous maps h and g, where h is positively homogeneous of degree \(\gamma \) (\(\ge 0\)) on C and \(g(x)=o(||x||^\gamma )\) as \(||x||\rightarrow \infty \) 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, \({\text {VI}}(f,K,q)\), of finding an \(x^*\in K\) such that \(\langle f(x^*)+q,x-x^*\rangle \ge 0\) for all \(x\in K.\) In this paper, we establish some results connecting the variational inequality problem \({\text {VI}}(f,K,q)\) and the cone complementarity problem \({\text {CP}}(f^{\infty },K^{\infty },0)\), where \(f^{\infty }:=h\) is the homogeneous part of f and \(K^{\infty }\) is the recession cone of K. We show, for example, that \({\text {VI}}(f,K,q)\) has a nonempty compact solution set for every q when zero is the only solution of \({\text {CP}}(f^{\infty },K^{\infty },0)\) and the (topological) index of the map \(x\mapsto x-\Pi _{K^{\infty }}(x-G(x))\) at the origin is nonzero, where G is a continuous extension of \(f^{\infty }\) 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 \({\mathcal {R}}^n\). 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 (Proc Am Math Soc 132:945–953, 2004) on the solvability of symmetric word equations to Euclidean Jordan algebras.

Keywords

Variational inequality problem Weakly homogeneous map Complementarity problem Degree Word equation 

Mathematics Subject Classification

90C33 15A24 17C20 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimoreUSA
  2. 2.Instituto de Ciencias de la IngenieríaUniversidad de O’HigginsRancaguaChile

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