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**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.

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.

Reviewer: Maxim Ivanov Todorov (Puebla)

### MSC:

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 |