(Kurdyka-Eojasiewicz property and exponent). We say that a proper closed function $h: \mathbb{X} \rightarrow \overline{\mathbb{R}}$ satisfies the Kurdyka-Eojasiewicz $(K L)$ property at $\hat{x} \in$ dom $\partial h$ if there are $a \in(0, \infty], a$ neighborhood $V$ of $\hat{x}$ and a continuous concave function $\varphi:[0, a) \rightarrow[0, \infty)$ with $\varphi(0)=0$ such that
(i) $\varphi$ is continuously differentiable on $(0, a)$ with $\varphi^{\prime}>0$ on $(0, a)$;
(ii) For any $x \in V$ with $h(\hat{x})<h(x)<h(\hat{x})+a$, it holds that $$ \varphi^{\prime}(h(x)-h(\hat{x})) \operatorname{dist}(0, \partial h(x)) \geq 1 $$
(Regular, defined by Clarke, in Optimization and Nonsmooth Analysis) $f$ is said to be regular at $x$ provided (i) For all $v,$ the usual one-sided directional derivative $f^{\prime}(x ; v)$ exists. (ii) For all $v, f^{\prime}(x ; v)=f^{\circ}(x ; v)$. where $f^{\circ}(x ; v)=\limsup _{y \rightarrow x \atop t \downarrow 0} \frac{f(y+t v)-f(y)}{t}$,$f^{\prime}(x ; v):=\lim _{t \downarrow 0} \frac{f(x+t v)-f(x)}{t}$.
If convexity of a function $f$ is not assumed, then KL property and regular are two weak assumptions of a function. Now I want to ask, which one of them is stronger? If the comparison is not proper, in which case should I use KL property, and in which case should I use regular property?
