Characterization of nonsmooth functions through their generalized gradients. (English) Zbl 0734.49005

Ellaia, R.; Hassouni, A.

Optimization 22, No. 3, 401-416 (1991).

The work deals with the question: how to characterize a given class of real-valued locally Lipschitz functions f(x) in terms of Clarke’s generalized gradient \(\partial f(x)\). Conditions on \(\partial f(x)\) necessary and sufficient for f(x) to be (i) quasi-convex and (ii) the difference of convex functions are established. The paper also contains a review of known conditions on \(\partial f(x)\) (obtained by R. T. Rockafellar and J. P. Vial) necessary and sufficient for f(x) to belong to the classes of (iii) convex functions, (iv) pointwise supremum of \(C^ k\)-functions, \(k\geq 1\) and(v) semi-smooth functions.

Reviewer: M.Yu.Kokurin (Yoshkar-Ola)

MSC:

49J52

Nonsmooth analysis

Keywords:

nonsmooth analysis; real-valued locally Lipschitz functions; Clarke’s generalized gradient; difference of convex functions

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