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Journal of Convex Analysis 26 (2019), No. 4, 1059--1070
Copyright Heldermann Verlag 2019



Gradients on Sets

Jan Mankau
Fakultšt Mathematik, Technische Universitšt, 01062 Dresden, Germany
jan.mankau@tu-dresden.de

Friedemann Schuricht
Fakultšt Mathematik, Technische Universitšt, 01062 Dresden, Germany
friedemann.schuricht@tu-dresden.de



[Abstract-pdf]

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

Keywords: Generalized gradient, set-valued gradient, generalized directional derivative, Lipschitz continuous function, optimal descent direction.

MSC: 46T20, 47H04, 49K99, 65K10.

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