Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies
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 th...
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| Ngôn ngữ: | English |
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Society for Industrial and Applied Mathematics
2021
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| Truy cập trực tuyến: | http://scholar.dlu.edu.vn/handle/123456789/500 |
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oai:scholar.dlu.edu.vn:123456789-5002022-02-10T22:56:32Z Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies Phạm, Tiến Sơn Local minimizers Lojasiewicz gradient inequality Optimality conditions Semialgebraic Sharp minimality Strong metric subregularity Tangencies 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]. 30 3 1777–1794 2021-08-23T08:09:07Z 2021-08-23T08:09:07Z 2020-07 Journal article Bài báo đăng trên tạp chí thuộc ISI, bao gồm book chapter http://scholar.dlu.edu.vn/handle/123456789/500 10.1137/19M1237466 en ISSN (print): 1052-6234 ISSN (online): 1095-7189 Society for Industrial and Applied Mathematics |
| institution |
Thư viện Trường Đại học Đà Lạt |
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Thư viện số |
| language |
English |
| topic |
Local minimizers Lojasiewicz gradient inequality Optimality conditions Semialgebraic Sharp minimality Strong metric subregularity Tangencies |
| spellingShingle |
Local minimizers Lojasiewicz gradient inequality Optimality conditions Semialgebraic Sharp minimality Strong metric subregularity Tangencies Phạm, Tiến Sơn Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| description |
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]. |
| format |
Journal article |
| author |
Phạm, Tiến Sơn |
| author_facet |
Phạm, Tiến Sơn |
| author_sort |
Phạm, Tiến Sơn |
| title |
Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| title_short |
Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| title_full |
Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| title_fullStr |
Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| title_full_unstemmed |
Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies |
| title_sort |
local minimizers of semi-algebraic functions from the viewpoint of tangencies |
| publisher |
Society for Industrial and Applied Mathematics |
| publishDate |
2021 |
| url |
http://scholar.dlu.edu.vn/handle/123456789/500 |
| _version_ |
1768305772153798656 |