Complementary Problems with Polynomial Data

Abstract

Given polynomial maps \(f, g \colon \mathbb {R}^{n} \to \mathbb {R}^{n}\), we consider the polynomial complementary problem of finding a vector \(x \in \mathbb {R}^{n}\) such that $$ f(x) \ge 0, \quad g(x) \ge 0, \quad \text{ and } \quad \langle f(x), g(x) \rangle = 0. $$ 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 explicit exponents. These strengthen and generalize some previously known results.