Existence theorems for optimal solutions in semi-algebraic optimization

TL;DR

This paper uses semi-algebraic geometry to establish conditions for the existence, boundedness, and optimality of solutions in semi-algebraic optimization problems, providing verifiable criteria and a formula for the optimal value.

Contribution

It introduces new verifiable conditions for solution existence and boundedness in semi-algebraic optimization, along with a computable optimal value formula.

Findings

Necessary and sufficient conditions for solution existence.

Criteria for boundedness from below and coercivity.

A computable formula for the optimal value.

Abstract

Consider the problem of minimizing a lower semi-continuous semi-algebraic function $f \colon \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ 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.