Clarke Subgradients of Stratifiable Functions

Abstract

We establish the following result: If the graph of a lower semicontinuous real-extended-valued function $f:\mathbb{R} ^{n}\rightarrow\mathbb{R}\cup\{+\infty\}$ admits a Whitney stratification (so in particular if $f$ is a semialgebraic function), then the norm of the gradient of $f$ at $x\in\mbox{dom\,}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain a Morse-Sard type of theorem as well as a nonsmooth extension of the Kurdyka-Lojasiewicz inequality for functions definable in an arbitrary o-minimal structure. It is worthwhile pointing out that, even in a smooth setting, this last result generalizes the one given in [K. Kurdyka, Ann. Inst. Fourier (Grenoble), 48 (1998), pp. 769-783] by removing the boundedness assumption on the domain of the function.