Local minimizers of semi-algebraic functions

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].



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