In-Depth Analysis of Linear Programming

By: F.P. Vasilyev, A.Y. Ivanitskiy

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Hardcover | 31 October 2001

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328 Pages

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Along with the traditional material concerning linear programming (the simplex method, the theory of duality, the dual simplex method), In-Depth Analysis of Linear Programming contains new results of research carried out by the authors.
For the first time, the criteria of stability (in the geometrical and algebraic forms) of the general linear programming problem are formulated and proved. New regularization methods based on the idea of extension of an admissible set are proposed for solving unstable (ill-posed) linear programming problems. In contrast to the well-known regularization methods, in the methods proposed in this book the initial unstable problem is replaced by a new stable auxiliary problem. This is also a linear programming problem, which can be solved by standard finite methods. In addition, the authors indicate the conditions imposed on the parameters of the auxiliary problem which guarantee its stability, and this circumstance advantageously distinguishes the regularization methods proposed in this book from the existing methods. In these existing methods, the stability of the auxiliary problem is usually only presupposed but is not explicitly investigated.
In this book, the traditional material contained in the first three chapters is expounded in much simpler terms than in the majority of books on linear programming, which makes it accessible to beginners as well as those more familiar with the area.

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Non-Fiction Mathematics Optimisation Linear Programming Computing & I.T.Computer Programming & Software Development Applied Mathematics Computer Science Economics Economic Theory & Philosophy Business & Management Operational Research

Preface
Introduction
Acknowledgments
Simplex Method	p. 1
Statement of the Problem	p. 1
Geometrical Interpretation. Extreme Points	p. 10
The Main Scheme of the Simplex Method	p. 19
Anticyclin	p. 45
Search for the Initial Extreme Point. Conditions for Solvability of a Canonical Problem	p. 58
The Main Theorems of Linear Programming	p. 79
Condition for Solvability of the General Problem	p. 79
The Duality Theorems	p. 82
The M-Method	p. 94
Other Theorems	p. 104
Evaluation of the Distance Between a Point and a Polyhedron (Hoffman Inequality)	p. 110
Dual Simplex Method	p. 119
Description of the Method	p. 119
Interpreting the Method for a Dual Problem	p. 135
Choice of the Initial Point	p. 146
Dual Interpretation of the Main Simplex Method	p. 161
Criterion of Stability	p. 167
Examples. Definitions	p. 167
The Necessary Condition for Stable Solvability	p. 171
Criteria of Boundedness of Polyhedrons	p. 174
Criteria of Stable Solvability	p. 182
Equivalence of Different Concepts of Stability	p. 192
Regularization Methods	p. 203
Stabilization Method	p. 203
The Method of Residual	p. 218
The Method of Quasisolutions	p. 222
Polynomial Methods in Linear Programming	p. 229
Problem Statement	p. 229
Khachiyan's Method	p. 240
Karmarkar's Method	p. 249
Nesterov's Method	p. 266
Notation	p. 295
References	p. 299
Index	p. 311
Table of Contents provided by Blackwell. All Rights Reserved.

In-Depth Analysis of Linear Programming

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