Local Minimizers Of Semi-Algebraic Functions From The Viewpoint Of Tangencies

Abstract

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].