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Zbl 0734.49005
Ellaia, R.; Hassouni, A.
Characterization of nonsmooth functions through their generalized gradients.
(English)
[J] Optimization 22, No.3, 401-416 (1991). ISSN 0233-1934; ISSN 1029-4945/e

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 $C\sp k$-functions, $k\ge 1$ and(v) semi-smooth functions.
[M.Yu.Kokurin (Yoshkar-Ola)]
MSC 2000:
*49J52 Nonsmooth analysis (other weak concepts of optimality)

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

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