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Title: Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies
Authors: Phạm, Tiến Sơn 
Keywords: Local minimizers;Lojasiewicz gradient inequality;Optimality conditions;Semialgebraic;Sharp minimality;Strong metric subregularity;Tangencies
Issue Date: 2020-07
Publisher: Society for Industrial and Applied Mathematics
Volume: 30
Issue: 3
Pages: 1777–1794
Consider a semialgebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so-called tangency variety of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of $f,$ we define a “tangency exponent” $\alpha_* > 0$ so that for any $\alpha \in \mathbb{R}$ the following four conditions are always equivalent: (i) the inequality $\alpha \ge \alpha_*$ holds, (ii) the point $\bar{x}$ is an $\alpha$th order sharp local minimizer of $f$, (iii) the limiting subdifferential $\partial f$ of $f$ is $(\alpha - 1)$th order strongly metrically subregular at $\bar{x}$ for 0, and (iv) the function $f$ satisfies the Łojaseiwcz gradient inequality at $\bar{x}$ with the exponent $1 - \frac{1}{\alpha}.$ Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe [Math. Program. Ser. A, 153 (2015), pp. 635--653].
DOI: 10.1137/19M1237466
Type: Bài báo đăng trên tạp chí thuộc ISI, bao gồm book chapter
Appears in Collections:Tạp chí (Khoa Toán - Tin học)

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