Nonsmooth Optimization and Related Topics pp 133-154 | Cite as

# On Cone Approximations and Generalized Directional Derivatives

Chapter

## Abstract

By the classical notion of differentiability necessary optimality conditions for smooth problems can be formulated as stationarity conditions. But often optimization problems connected with real life problems have a “nondifferentiable” structure. Hence it is useful to develop appropriate notions of generalized differentiability in order to obtain analogous optimality conditions in nondifferentiable case, too. The concepts of the directional derivative and the subdifferential of a convex function were used with advantage for treating convex optimization problems. Since more than ten years much effort was made to establish similar concepts in the nonconvex case. Several trends can be distinguished, for example modifications of the directional derivative given by Clarke and many other authors. In accordance with such investigations, in Refs.[4, 5, 16] an axiomatic approach was given for constructing generalized directional derivatives of arbitrary extended real-valued functionals. The basic idea is the fact that the epigraphs of the different directional derivatives of a function \(f:X \to \overline R\) ( is determined uniquely by that cone.

*X*a linear topological space) can be considered as cone approximations of the epigraph epi*f*of*f*. Conversely, so called*K*-directional derivatives can be introduced in such a way that epi*f*is locally approximated by a cone*K*(epi*f*, (*x*,*f*(*x*))) at the point (*x*,*f*(*x*)), where a positively homogeneous functional \({f^K}(x, \cdot ):X \to \overline R\) according$${f^K}(x,y):\inf \left\{ {\xi \in R|(y,\xi ) \in K(epif,(x)))} \right\}$$

## Keywords

Convex Cone Directional Derivative Generalize Derivative Tangent Cone Nonsmooth Optimization
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