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Characterization of nonsmooth functions through their generalized gradients. (English) Zbl 0734.49005
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 $\partial f\left(x\right)$. Conditions on $\partial f\left(x\right)$ 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 $\partial f\left(x\right)$ (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, $k\ge 1$ and(v) semi-smooth functions.

##### MSC:
 49J52 Nonsmooth analysis (other weak concepts of optimality)