Mordukhovich, Boris S. Generalized differential calculus for nonsmooth and set-valued mappings. (English) Zbl 0807.49016 J. Math. Anal. Appl. 183, No. 1, 250-288 (1994). Some generalized differentiability concepts for multifunctions and nonsmooth mappings in \(\mathbb{R}^ n\) are studied. In this paper the most important one is the so-called coderivative of multifunctions introduced earlier [the author, Dokl. Akad. Nauk SSSR 254, 1072-1076 (1980; Zbl 0491.49011)] using the normal cone to the graph in the sense given by the author in Prikl. Mat. Mekh. 40, 1014-1023 (1976; Zbl 0362.49017). This normal cone is not convex. The corresponding nonconvex coderivative appears to be useful in control theory to obtain necessary optimality conditions in differential inclusions. In this paper a rich calculus for the coderivative and related subdifferential constructions is developed using a variational approach, namely an extremal generalization of the separability theorem for nonconvex sets. The proof of this result is based on a special smooth approximation procedure using proximal points (the so-called metric approximation method). Reviewer: V.Arnautu (Iaşi) Cited in 4 ReviewsCited in 119 Documents MSC: 49J52 Nonsmooth analysis 49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000) Keywords:set-valued mappings; generalized differentiability; multifunctions; nonsmooth mappings; coderivative; subdifferential Citations:Zbl 0491.49011; Zbl 0362.49017 PDF BibTeX XML Cite \textit{B. S. Mordukhovich}, J. Math. Anal. Appl. 183, No. 1, 250--288 (1994; Zbl 0807.49016) Full Text: DOI OpenURL