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Tangencies and Polynomial Optimization

Tien-Son PHAM (sonpt***at***dlu.edu.vn)

Abstract: Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ 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 $S.$

Keywords: Boundedness, Coercivity, Compactness, Critical points, Existence of minimizers, Polynomial, Semi-Algebraic, Stability, Sub-levels, Tangencies

Category 1: Global Optimization (Theory )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation:

Download: [PDF]

Entry Submitted: 02/16/2019
Entry Accepted: 02/17/2019
Entry Last Modified: 03/09/2019

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