# Linear Programming

## A Modern Integrated Analysis

Book

Part of the International Series in Operations Research & Management Science book series (ISOR, volume 1)

1. Front Matter
Pages i-xiii
2. Romesh Saigal
Pages 1-5
3. Romesh Saigal
Pages 7-65
4. Romesh Saigal
Pages 67-83
5. Romesh Saigal
Pages 85-110
6. Romesh Saigal
Pages 111-264
7. Romesh Saigal
Pages 265-305
8. Back Matter
Pages 307-342

### Introduction

In Linear Programming: A Modern Integrated Analysis, both boundary (simplex) and interior point methods are derived from the complementary slackness theorem and, unlike most books, the duality theorem is derived from Farkas's Lemma, which is proved as a convex separation theorem. The tedium of the simplex method is thus avoided.
A new and inductive proof of Kantorovich's Theorem is offered, related to the convergence of Newton's method. Of the boundary methods, the book presents the (revised) primal and the dual simplex methods. An extensive discussion is given of the primal, dual and primal-dual affine scaling methods. In addition, the proof of the convergence under degeneracy, a bounded variable variant, and a super-linearly convergent variant of the primal affine scaling method are covered in one chapter. Polynomial barrier or path-following homotopy methods, and the projective transformation method are also covered in the interior point chapter. Besides the popular sparse Cholesky factorization and the conjugate gradient method, new methods are presented in a separate chapter on implementation. These methods use LQ factorization and iterative techniques.

### Keywords

Algebra Complementary Slackness Newton's method Optimality Conditions calculus duality linear algebra linear optimization programming transformation

#### Authors and affiliations

1. 1.Department of Industrial and Operations EngineeringThe University of MichiganAnn ArborUSA

### Bibliographic information

• Book Title Linear Programming
• Book Subtitle A Modern Integrated Analysis
• Authors Romesh Saigal
• Series Title International Series in Operations Research & Management Science
• DOI https://doi.org/10.1007/978-1-4615-2311-6
• Publisher Name Springer, Boston, MA
• eBook Packages
• Hardcover ISBN 978-0-7923-9622-2
• Softcover ISBN 978-1-4613-5977-7
• eBook ISBN 978-1-4615-2311-6
• Series ISSN 0884-8289
• Edition Number 1
• Number of Pages XIII, 342
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site

## Reviews

`I recommend this book to anyone desiring a deep understanding of the simplex method, interior-point methods, and the connections between them.'
Interfaces, 27:2 (1997)
The book is clearly written. ... It is highly recommended to anybody wishing to get a clear insight in the field and in the role that duality plays not only from a theoretical point of view but also in connection with algorithms.'
Optimization, 40 (1997)