Paper detail

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

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" $α_* > 0$ so that for any $α\in \mathbb{R}$ the following four conditions are always equivalent: (i) the inequality $α\ge α_*$ holds; (ii) the point $\bar{x}$ is an $α$th order sharp local minimizer of $f;$ (iii) the limiting subdifferential $\partial f$ of $f$ is $(α- 1)$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 $1 - \frac{1}α.$ Besides, we also present a counterexample to a conjecture posed by Drusvyatskiy and Ioffe [Math. Program. Ser. A, 153(2):635--653, 2015].