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Generalized convexity, nonsmooth variational inequalities, and nonsmooth optimization. (English) Zbl 1276.49001

Boca Raton, FL: CRC Press (ISBN 978-1-4398-6820-1/hbk; 978-1-4398-6821-8/ebook). xv, 280 p. (2014).
The book presents the concepts of convexity, monotonicity and variational inequalities with their generalizations. It is divided into two parts. The first part deals with generalized convexity and monotonicity for differentiable and nondifferentiable functions. The second part of the book deals with variational inequalities and optimization problems.
The book is divided into 8 chapters. Chapter 1 presents elements of convex analysis such as convex sets, convex functions, some generalizations of convexity, optimality criteria and notions of subgradients and subdifferentials. Chapter 2 is devoted to the concept of differentiability and its generalizations. We find here Gâteaux derivative, Dini and Dini-Hadamard derivatives, Clarke derivative as well as Clarke and Dini subdifferentials. Chapter 3 deals with nonsmooth convexity, in particular some generalizations of convexity in terms of bifunctions. Chapter 4 presents concepts of generalized monotonicity and its relations with generalized convexity. In particular, the notion of generalized monotonicity in terms of bifunctions and with respect to set-valued maps are introduced.
Further four chapters form the second part of the book. In chapter 5 we find variational inequalities, basic existence and uniqueness results and methods of finding solutions. Chapter 6 describes nonsmooth variational inequalities and their relations to optimization problems. In chapter 7 we find some characterizations of solution sets of optimization problems and nonsmooth variational inequalities, in particular in terms of pseudoaffine bifunctions and Lagrange multipliers. Finally, chapter 8 deals with nonsmooth generalized variational inequalities and related optimization problems. We find here the notions of gap functions, Clarke subdifferential and generalized pseudolinear objective functions.
The book contains a lot of examples illustrating the introduced notions. It also contains some nice diagrams showing the relations between different notions of convexity or monotonicity.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J40 Variational inequalities
49J52 Nonsmooth analysis
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