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Nonsmooth analysis. (English) Zbl 1120.49001

Universitext. Berlin: Springer (ISBN 978-3-540-71332-6/pbk). xii, 373 p. (2007).
The main idea of the presented monograph is to deal with extremum problems connected with nondifferentiable data. The text is divided into 13 chapters with an Appendix and 229 references. Each chapter ends with recommended references and exercises. The book starts with simpler problems and methods, gradually passes to more difficult tasks. The first two chapters deal with convex sets and convex functionals. Various types of derivatives and relations between them are investigated in the third chapter. The \(\beta\)-differentiability with respect to a bornology \(\beta\) is considered too. The next chapter deals with subdifferentials of convex functionals. In order to study nonconvex problems special types of subdifferentials and variational principles are explained. Multifunctions and their calculus are considered in the tenth chapter. Optimality conditions are derived both for convex and nonconvex problems. The last chapter is devoted mainly to the results achieved by B. S. Mordukhovich in recent years. It concern Mordukhovich normals, subdifferentials and their calculus and ends with applications to multiobjective optimization. The text contains a big amount of latest results achieved in nonsmooth analysis together with applications in optimization. It can be recommended both to graduate students and the researchers in applied mathematics and optimization.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J52 Nonsmooth analysis
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