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arXiv · 2308.05349

Existence theorems for optimal solutions in semi-algebraic optimization

Jae Hyoung Lee·Gue Myung Lee·Tien Son Pham
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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.

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Jae Hyoung Lee, Gue Myung Lee, Tien Son Pham. 2023-08-10. Existence theorems for optimal solutions in semi-algebraic optimization. https://arxiv.org/abs/2308.05349

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