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Clarke subgradients of stratifiable functions

J. Bolte·A. Daniilidis·A. S. Lewis·M. Shiota
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Abstract

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-Łojasiewicz inequality for functions definable in an arbitrary o-minimal structure.

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J. Bolte, A. Daniilidis, A. S. Lewis, M. Shiota. 2006-01-22. Clarke subgradients of stratifiable functions. https://arxiv.org/abs/math/0601530

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