Optimization and nonsmooth analysis. Transl. from the English by Yu. S. Ledyaev. Transl. ed. by V. I. Blagodatskikh. (Оптимизация и негладкий анализ.) (Russian) Zbl 0662.49001

Moskva: Nauka. 280 p. R. 2.40 (1988).
Translation from the English original (1983), reviewed in Zbl 0582.49001.
The book is devoted to nonsmooth analysis and optimization problems. In the author’s work the notions of gradient and subdifferential are introduced for arbitrary locally Lipschitz functions. The rather extensive theory of subdifferential calculus is quoted in this book.
The book consists of seven chapters, the first being an introduction. The theory of generalized gradients is discussed in detail in chapter two which considers locally Lipschitz functions defined on Banach spaces.
The next two chapters pay special attention to the problem of dynamic optimization: the first considers the problem of optimal control for differential inclusions, while the second studies Bolza’s generalized problem.
Chapter five presents results from the theory of optimal control, i.e., the problem of existence of optimal trajectories, necessary and sufficient conditions for optimality on the problems of controllability and sensitivity. The following chapter considers the optimization problem as a general problem arising in mathematical programming. Interesting relationships are described here between Lagrange multipliers and the necessary conditions, the value function, stability and sensitivity of the problems.
The last chapter deals with some problems not directly related to optimization such as obtaining the theorem for reversible and implicit functions, and Aumann’s theorem. The classical theory of variational equations is considered. Ekeland’s theorem is also discussed and its application to nonsmooth analysis is illustrated.


49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49J52 Nonsmooth analysis
49K15 Optimality conditions for problems involving ordinary differential equations
26B05 Continuity and differentiation questions
26E15 Calculus of functions on infinite-dimensional spaces
46G05 Derivatives of functions in infinite-dimensional spaces
49J15 Existence theories for optimal control problems involving ordinary differential equations
34A60 Ordinary differential inclusions
93B05 Controllability
93B03 Attainable sets, reachability
49K40 Sensitivity, stability, well-posedness
90C30 Nonlinear programming


Zbl 0582.49001