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A course in optimization methods. (Курс методов оптимизации.) (Russian) Zbl 0602.90091

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 328 p. R. 1.80 (1986).
This is a textbook for university students in applied mathematics and economic cybernetics. It includes traditional topics of optimization theory and algorithms.
In Chapter 1 (Introduction to optimization) the basic definitions are given as well as the classification of optimization problems. The Chapter 2 on univariate methods includes the classical algorithms. In addition global minimization methods of Lipschitz functions and optimal algorithms are discussed. The next chapters ”Foundations of convex analysis” and ”Theory of necessary and sufficient conditions of optimality” contain the usual material of textbooks on these subjects. In Chapter 5 the unconstrained methods are considered, first of all, gradient and Newton ones. Methods of variable metrics, conjugate directions and search are briefly introduced. One of the main chapters is the sixth on numerical methods for constrained problems. Linear and quadratic programming problems are analyzed here. The following methods for the general nonlinear programming problem are described: projected gradients, reduced gradients, penalties, parametrization and linearization. The examples of discrete optimization problems and ideas of the algorithms are introduced in the seventh Chapter. In last Chapter ”Elements of optimal control” the problem is introduced, the proof of the Pontryagin maximum principle is presented and examples of its application are demonstrated.
The text is written in strict mathematical style and aims to introduce the reader to the mathematical background of the subject.

MSC:

90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
90C05 Linear programming
90C25 Convex programming
90C30 Nonlinear programming
90C10 Integer programming
49M37 Numerical methods based on nonlinear programming
90C55 Methods of successive quadratic programming type
65K05 Numerical mathematical programming methods
65K10 Numerical optimization and variational techniques