Linear programming. 2nd ed.
(Линейное программирование.)

*(Russian)*Zbl 1200.90003
Seriya “XX Vek. Matematika i Mekhanika”. Moscow: Faktorial Press (ISBN 5-88688-062-3). 348 p. (2003).

This is the second, extended edition of this textbook. The authors’ aim is to provide engineering and business students as well as mathematical optimization theorists with a textbook containing a mathematically rigorous explanation of the theory and methods of linear programming, which requires only minimum knowledge of mathematical analysis and linear algebra (e.g., even the usage of separation theorems is avoided).

The book is divided into 7 chapters. Chapter 1 concerns the formulation of linear programming problems and continues with a basic description of the simplex method for solving these problems. The description includes dealing with degeneracy in linear programming problems and overcoming the cycling of the original basic method by making use of anti-cycling procedures. The main theoretical properties of linear programming problems are included in Chapter 2 and the duality theory is contained in the next chapter. Chapter 4 is devoted to the formulation and solution of a linear programming problem with a special structure, namely the transportation problem. Three methods for solving this problem are considered: the north-west corner method, the potential method, and the crossing method. Chapter 5 deals with the stability of the general form of the linear programming problem. The authors define and compare various concepts of stability. Chapter 6 includes three regularization procedures designed for incorrectly posed linear programming problems, namely the stabilization method, the method of residuals, and the method of quasi-solutions. The concluding Chapter 7 is devoted to polynomial methods for solving linear programming problems (methods by Chačian, Karmarkar, and Nesterov).

The book is divided into 7 chapters. Chapter 1 concerns the formulation of linear programming problems and continues with a basic description of the simplex method for solving these problems. The description includes dealing with degeneracy in linear programming problems and overcoming the cycling of the original basic method by making use of anti-cycling procedures. The main theoretical properties of linear programming problems are included in Chapter 2 and the duality theory is contained in the next chapter. Chapter 4 is devoted to the formulation and solution of a linear programming problem with a special structure, namely the transportation problem. Three methods for solving this problem are considered: the north-west corner method, the potential method, and the crossing method. Chapter 5 deals with the stability of the general form of the linear programming problem. The authors define and compare various concepts of stability. Chapter 6 includes three regularization procedures designed for incorrectly posed linear programming problems, namely the stabilization method, the method of residuals, and the method of quasi-solutions. The concluding Chapter 7 is devoted to polynomial methods for solving linear programming problems (methods by Chačian, Karmarkar, and Nesterov).

Reviewer: K. Zimmermann (MR 2094318)