Mathematical programming. (Matematicheskoe programmirovanie) (Q2720885)

[For a review of the English ed. (Moscow, Mir) (1989) see Zbl 0713.90075).]NEWLINENEWLINENEWLINEThis textbook on the theory and basic methods of mathematical programming is based on the author's lecturing activities at the Moscow State University. The book is divided into 11 chapters.NEWLINENEWLINENEWLINEChapter 1 defines the subject of mathematical programming, determines its places within operations research and brings some technical-economic problems, the mathematical formulation of which leads to the necessity to solve mathematical programming problems. The necessary mathematical background for treating mathematical programming problems is contained in Chapter 2. In this chapter, we find the elements of convex analysis, especially the properties of convex sets and functions. Chapter 3 is devoted to the investigation of necessary optimality conditions for local minima of general optimization problems and necessary and sufficient conditions for convex optimization).NEWLINENEWLINENEWLINEThe basic theoretical results concerning linear programming and basic solution methods for LP problems are described in Chapter 4 and 5. Widely used penalty function methods are studied in Chapter 6. Since the initial information, which is used to formulate optimization problems is not exact, it is important to point out such classes of optimization problems and solution methods, which are not too sensitive to small changes in the initial informationNEWLINENEWLINENEWLINETheoretical problems connected with this task are studied in Chapter 7.NEWLINENEWLINENEWLINEThe necessity to solve a one-dimensional optimization problem occurs very often as a part of general optimization algorithms solving \(n\)-dimensional problems so that the choice of appropriate algorithms for one-dimensional subproblems can substantly influence the effectivity of the original main algorithm. For this reason, Chapter 8 is devoted to the investigation of one-dimensional optimization procedures. Stabilising properaties of methods using gradients are contained in Chapter 9. NEWLINENEWLINENEWLINEChapter 10, the author includes solution methods for optimization problems with constraints: projected gradient and conditional gradient methods, the method of feasible directions and stochastic descent methods. Methods based on the usage of augmented Lagrangians are described in the last chapter.NEWLINENEWLINENEWLINEThree appendices in the end of the book deal with the finiteness of one numerical method, estimation of the convergence rate of a coordinate descent method and degeneracy optimization problems.