Linear Programming Vasek Chvatal

Number 1 linear programming course. Learn the simplex method, duality and sensitivity analysis for linear programs

Today many economists, engineers and mathematicians are familiar with linear programming and are able to apply it. This is owing to the following facts: during the last 25 years efficient methods have been developed; at the same time sufficient computer capacity became available; finally, in many different fields, linear programs have turned out to be appropriate models for solving practical problems. However, to apply the theory and the methods of linear programming, it is required that the data determining a linear program be fixed known numbers.

To this reviewer’s knowledge, this is the first book accessible to the upper division undergraduate or beginning graduate student that surveys linear programming from the Simplex Method…via the Ellipsoid algorithm to Karmarkar’s algorithm. Moreover, its point of view is algorithmic and thus it provides both a history and a case history of work in complexity theory. The presentation is admirable; Karloff's style is informal (even humorous at times) without sacrificing anything necessary for understanding. Diagrams (including horizontal brackets that group terms) aid in providing clarity. The end-of-chapter notes are helpful…Recommended highly for acquisition, since it is not only a textbook, but can also be used for independent reading and study.

The main theorem of Linear Programming Duality, relating a "pri­ mal" Linear Programming problem to its "dual" and vice versa, can be seen as a statement about sign patterns of vectors in complemen­ tary subspaces of Rn. This observation, first made by R.T. Rockafellar in the late six­ ties, led to the introduction of certain systems of sign vectors, called "oriented matroids". Indeed, when oriented matroids came into being in the early seventies, one of the main issues was to study the fun­ damental principles underlying Linear Progra.mrning Duality in this abstract setting.

This book presents the necessary and essential backgrounds of fuzzy set theory and linear programming, particularly a broad range of common Fuzzy Linear Programming (FLP) models and related, convenient solution techniques. These models and methods belong to three common classes of fuzzy linear programming, namely: (i) FLP problems in which all coefficients are fuzzy numbers, (ii) FLP problems in which the right-hand-side vectors and the decision variables are fuzzy numbers, and (iii) FLP problems in which the cost coefficients, the right-hand-side vectors and the decision variables are fuzzy numbers.

Linear programming is a relatively modern branch of Mathe­ matics, which is a result of the more scientific approach to management and planning of the post-war era. The purpose of this book is to present a mathematical theory of the subject, whilst emphasising the applications and the techniques of solution. An introduction to the theory of games is given in chapter five and the relationship between matrix games and linear programmes is established. The book assumes that the reader is familiar with matrix algebra and the background knowledge required is covered in the book, Linear Equations by P.M. Cohn, of this series. In fact the notation used in this text conforms with that intro­ duced by Cohn. The book is based on a course of about 18 lectures given to Mathematics and Physics undergraduates. Several examples are worked out in the text and each chapter is followed by a set of examples. I am grateful to my husband for many valuable suggestions and advice, and also to Professor W. Ledermann, for encourag­ ing me to write this book.

Linear Programming provides an in-depth look at simplex based as well as the more recent interior point techniques for solving linear programming problems. Starting with a review of the mathematical underpinnings of these approaches, the text provides details of the primal and dual simplex methods with the primal-dual, composite, and steepest edge simplex algorithms.

With emphasis on computation, this book is a real breakthrough in the field of LP. In addition to conventional topics, such as the simplex method, duality, and interior-point methods, all deduced in a fresh and clear manner, it introduces the state of the art by highlighting brand-new and advanced results, including efficient pivot rules, Phase-I approaches, reduced simplex methods, deficient-basis methods, face methods, and pivotal interior-point methods. In particular, it covers the determination of the optimal solution set, feasible-point simplex method, decomposition principle for solving large-scale problems, controlled-branch method based on generalized reduced simplex framework for solving integer LP problems.