Abstract

For a locally Lipschitz continuous function f ⁣:X→Rf\colon X\to\mathbb{R} the generalized gradient ∂f(x)\partial f(x) of Clarke is used to develop some (set-valued) gradient on a set A⊂XA\subset X. Existence, uniqueness and some approximation are considered for optimal descent directions on set AA. The results serve as basis for nonsmooth numerical descent algorithms that can be found in subsequent papers.

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Keywords

  • Generalized gradient
  • set-valued gradient
  • generalized directional derivative
  • Lipschitz continuous function
  • optimal descent direction

Mathematics Subject Classification

46T20, 47H04, 49K99, 65K10

References

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Suggested citation

J. Mankau, F. Schuricht. “Gradients on Sets.” Journal of Convex Analysis 26 (2019), No. 4, 1059–1070. https://doi.org/10.68381/jca26057

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.