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TienSon PHAM (sonptdlu.edu.vn) Abstract: Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semialgebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the socalled {\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, SemiAlgebraic, Stability, Sublevels, Tangencies Category 1: Global Optimization (Theory ) Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization ) Citation: Download: [PDF] Entry Submitted: 02/16/2019 Modify/Update this entry  
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