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Clarke subgradients for directionally Lipschitzian stratifiable functions

Dmitriy Drusvyatskiy (dd379***at***cornell.edu)
Alexander Ioffe (ioffe***at***math.technion.ac.il)
Adrian Lewis (aslewis***at***orie.cornell.edu)

Abstract: Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $\R^n$ have a limiting subdifferential graph of uniform local dimension $n$.

Keywords: Nonsmooth analysis; subdifferential; gradient sampling; stratification; semi-algebraic; dimension

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 11/15/2012
Entry Accepted: 11/15/2012
Entry Last Modified: 12/03/2012

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