Tangencies and Polynomial Optimization

Abstract

Given a polynomial function $f:{\mathbb{R}}^n\to \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S\subset {\mathbb{R}}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The $f$ attains its infimum on $S;$ (iii) The sublevel set $\{x\in S|f(x)\le \lambda \}$ for $\lambda \in \mathbb{R}$ is compact; (iv) The $f$ is coercive on $S.$ Besides, we also provide some stability criteria for boundedness and coercivity of $f$ on $S.$

a minor change in the introduction