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Subdifferentials: Theory and applications. Transl. from the Russian. (English) Zbl 0832.49012
Mathematics and its Applications (Dordrecht). 323. Dordrecht: Kluwer Academic Publishers. ix, 398 p. $ 189.00; Dfl. 295.00; £121.00 (1995).
Already in 1987 the authors presented a book entitled “Subdifferential calculus” (in Russian) in which the most important definitions and principles of convex analysis are treated in a modern setting. The four chapters contain a detailed description of convex sets and convex mappings (Chapter 1), a geometrical and topological treatment of subdifferentials for convex functions (Chapter 2 and 3) and the presentation of subdifferential calculus rules using the Young-Fenchel transformation (Chapter 4). For a detailed specification we refer to the review in Zbl 0655.49001.
Five years later, in 1992, the same authors published the book “Subdifferentials. Theory and applications” in which the four chapters of the former book are extended by a chapter about the application of the subdifferential theory in mathematical optimization (see Zbl 0760.49012).
The present book is a translation of the last mentioned book in the English language. In addition to the mentioned five chapters, the authors provide a sixth chapter on local convex approximations. Here an extensive classification of modern cone approximations of sets used in nonsmooth optimization is given. Starting with different definitions of the well- known Clarke tangent cone the authors discuss many other cones which are described by elementary set operations, by generalized sequences and by use of quantifiers, respectively. Close connections to Kuratowski limits of set-valued mappings and to generalized directional derivatives of functions are pointed out. Especially by transposition of the quantifiers new interesting constructions can be stated. It is demonstrated that the representation of all notions can be shortened by use of modern principles of nonstandard analysis.
After each chapter some comments about the described theory and other related topics are enclosed. Finally, a large bibliographical list at the end of the book refers to further literature on the subject.

49J52 Nonsmooth analysis
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
90C48 Programming in abstract spaces
90C25 Convex programming
90C29 Multi-objective and goal programming
47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices