Optimization. Introduction to mathematical theory and methods (Q1738350)

This textbook is devoted to the theory and methods of optimization in finite-dimensional spaces. The book is divided into four parts. The first part describes methods of linear programming. The authors consider the simplex method and various variants of interior-point methods. Special attention is devoted to applications of linear programming to transport and transshipment problems as well as to usage of linear programming in solving some network problems. The second part of the book contains methods for solving nonlinear optimization problems without constraints. The authors describe one-dimensional search methods for functions depending on one variable and, further, for functions depending on more variables, methods based on the usage of gradients are presented (steepest descent, conjugate gradients, trust region and Newton-Gauss methods). The third part is devoted to the theoretical background of nonlinear optimization for both convex and general problems (separation theorems, optimality conditions, Kuhn-Tucker theorem). The theoretical results of the third part are used in the fourth part, in which the authors explain various approaches to constrained nonlinear optimization problems (methods using projected gradient, penalty or barrier functions, primal-dual methods). Special chapters contain methods of semidefinite programming and direct search for functions with more variables. Convergence of the methods is studied. Special attention is devoted to applications of nonlinear optimization both to continuous problems of other areas of mathematics and to solution some discrete combinatorial problems. For the first edition, see [Optimierung. Berlin: Springer (2004; Zbl 1029.90002)].