Existence theorems for optimal solutions in semi-algebraic optimization

Abstract

Consider the problem of minimizing a lower semi-continuous semi-algebraic function $f:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ on an unbounded closed semi-algebraic set $S\subset {\mathbb{R}}^n.$ Employing adequate tools of semi-algebraic geometry, we first establish some properties of the tangency variety of the restriction of $f$ on $S.$ Then we derive verifiable necessary and sufficient conditions for the existence of optimal solutions of the problem as well as the boundedness from below and coercivity of the restriction of $f$ on $S.$ We also present a computable formula for the optimal value of the problem.