Characterization of subgradients. I (Q1116080)

The aim of the paper is to study relations between the generalized gradients of \textit{F. H. Clarke} [Trans. Am. Math. Soc. 205, 247-262 (1975; Zbl 0307.26012)] and the \(\phi_ 2\)-subgradients and \(Q^ c\)- subgradients introduced by \textit{S. Dolecki} and \textit{S. Kurcyusz} [SIAM J. Control Optimization 16, 277-300 (1978; Zbl 0397.46013)]. It is shown that, for a locally Lipschitzian function f: \({\mathbb{R}}^ n\to {\mathbb{R}}\), the existence of \(\phi_ 2\)-subgradients implies the existence of \(Q^ c\)-subgradients within a sufficiently small neighbourhood of a given point. Furthermore, the Clarke generalized gradient of f can be characterized as the convex closure of the derivatives of the \(Q^ c\)- subgradients.