Complementary problems with polynomial data

Abstract

Given polynomial maps $f,g:{\mathbb{R}}^n\to {\mathbb{R}}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x\in {\mathbb{R}}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad \textrm{ and } \quad \langle f(x), g(x) \rangle \ = \ 0. \end{equation*} In this paper, we present various properties on the solution set of the problem, including genericity, nonemptiness, compactness, uniqueness as well as error bounds with exponents explicitly determined. These strengthen and generalize some previously known results, and hence broaden the boundary knowledge of nonlinear complementarity problems as well.