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Minimization methods for non-differentiable functions. Transl. from the Russian by K. C. Kiwiel and A. Ruszczyński. (English) Zbl 0561.90058
Springer Series in Computational Mathematics, 3. Berlin etc.: Springer- Verlag. VIII, 162 p. DM 84.00 (1985).
This monograph treats numerical methods for minimizing nonsmooth functions, especially convex functions. Contents: Chapter 1: Special classes of nondifferentiable functions and generalizations of the concept of the gradient; Chapter 2: The subgradient method; Chapter 3: Gradient- type methods with dilatation; Chapter 4: Applications of methods for nonsmooth optimization to the solution of mathematical programming problems.
Chapters 1 and 2 are elementary; we note in Chapter 1 a paragraph devoted to ”almost differentiable functions”, a class of functions which is precursory to the ones encountered in modern nondifferentiable optimization. Chapter 3 (45 p.) is the core of the book; it reflects the thinkings and the works of the author on the so-called ”space dilatation methods”. In Chapter 4 (56 p.) various applications of subgradient methods are given: use in decomposition methods, solving distribution or production-transportation problems, solving nonlinear minimax problems, etc. The monograph ends with a list of references, the major part of them (70 %) are authors from the East. Additional references, not referred to in the book, correct somewhat this bias.
Reviewer: J.B.Hiriart-Urruty

MSC:
90C25 Convex programming
90C55 Methods of successive quadratic programming type
49M37 Numerical methods based on nonlinear programming
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming