  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 $\alpha$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 \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 [Math. Program. Ser. A, 153(2):635--653, 2015].

Optimization and Control