Local minimizers of semi-algebraic functions from the viewpoint of tangencies

Abstract

Consider a semi-algebraic function $f:{\mathbb{R}}^n\to \mathbb{R},$ 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" ${\alpha }_{\ast }>0$ so that for any $\alpha \in \mathbb{R}$ the following four conditions are always equivalent: (i) the inequality $\alpha \ge {\alpha }_{\ast }$ 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(2):635--653, 2015].