Convex analysis and nonlinear optimization. Theory and examples
The aim of the book is to provide the first-year graduate students with an appropriate textbook of convex analysis and nonlinear optimization. The book is divided into 10 chapters. The first two chapters are devoted to necessary theoretical background as the introduction of the Euclidean space, concept of symmetric matrices, characterization of local minimizers by optimality conditions, some basic results from the convex analysis (e.g. separation of convex sets, theorems of alternatives) and properties of functions, which are expressed as the maximum of a finite number of continuous convex functions (so-called max-functions). Chapter 3 deals with the Fenchel duality. Some more advanced results from the convex analysis (e.g. Fenchel biconjugation, Lagrange duality) are contained in Chapter 4 and some consequences for special cases are considered in Chapter 5. Various types of generalized derivatives and their usage in nonsmooth optimization are investigated in Chapters 6 and 7. The concept of fixed points and some fixed point theorems (Brower, Kakutani-Fan) as well as some results concerning variational inequalities are presented in Chapter 8. The final explanatory chapter illustrates some of the challenges arising in infinite dimensions. The explanations are completed at the end with Chapter 10 containing list of results and notation. The text of the book consists of short selfcontained sections, each followed by exercises. The book is appropriate both as a class text and for self-study.
scientific article; zbMATH DE number 1505640