- Local minimizers of semi-algebraic functions Tien-Son PHAM(sonptdlu.edu.vn) Abstract: Consider a semi-algebraic 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 {\em 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 $\alpha$-order sharp local minimizer of $f;$ (iii) the limiting subdifferential $\partial f$ of $f$ is $(\alpha - 1)$-order strongly metrically subregular at $\bar{x}$ for $0;$ and (iv) the function $f$ satisfies the \L ojaseiwcz 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 in [Math. Program. Ser. A, 153(2):635--653, 2015]. Keywords: Local minimizers, \L ojasiewicz gradient inequality, Optimality conditions, Semi-algebraic, Sharp minimality, Strong metric subregularity, Tangencies Category 1: Nonlinear Optimization (Unconstrained Optimization ) Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: Download: [PDF]Entry Submitted: 01/07/2019Entry Accepted: 01/07/2019Entry Last Modified: 01/07/2019Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.