Clarke Subgradients of Stratifiable Functions 论文

摘要

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–Łojasiewicz 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.