Computational Techniques of the Simplex Method

ISBN-13: 9781402073328 / Angielski / Twarda / 2002 / 325 str.

Istvan Maros

Opis

Linear Programming (LP) is perhaps the most frequently used optimization technique. One of the reasons for its wide use is that very powerful solution algorithms exist for linear optimization. Computer programs based on either the simplex or interior point methods are capable of solving very large-scale problems with high reliability and within reasonable time. Model builders are aware of this and often try to formulate real-life problems within this framework to ensure they can be solved efficiently. It is also true that many real-life optimization problems can be formulated as truly linear models and also many others can well be approximated by linearization. The two main methods for solving LP problems are the variants of the simplex method and the interior point methods (IPMs). It turns out that both variants have their role in solving different problems. It has been recognized that, since the introduction of the IPMs, the efficiency of simplex based solvers has increased by two orders of magnitude. This increased efficiency can be attributed to the following: (1) theoretical developments in the underlying algorithms, (2) inclusion of results of computer science, (3) using the principles of software engineering, and (4) taking into account the state-of-the-art in computer technology.
Theoretically correct algorithms can be implemented in many different ways, but the performance is dependent on how the implementation is done. The success is based on the proper synthesis of the above mentioned (1-4) components. Computational Techniques of the Simplex Method is a systematic treatment focused on the computational issues of the simplex method. It provides a comprehensive coverage of the most important and successful algorithmic and implementation techniques of the simplex method. It is a unique source of essential, never discussed details of algorithmic elements and their implementation. On the basis of the book the reader will be able to create a highly advanced implementation of the simplex method which, in turn, can be used directly or as a building block in other solution algorithms.

Recenzje

"This is really a nice, long awaited book...it covers all important aspects and techniques needed for an efficient, robust implementation of the simplex method...it deserves a place on the bookshelf of every OR, optimization professional, it could be marketed as "handbook of computational techniques"."
(Tamás Terlaky, Dept. of Computing and Software, McMaster University, Hamilton, ON, Canada)
"The book will be of great interest to people developing advanced LP solver codes customized for special purposes, and solvers for integer programming and combinatorial optimization, and also for researchers working in these areas. It is an ideal textbook for graduate courses in computational mathematical programming. It deserves to be in the personal library of software engineers dealing with numerical computation involving linear constraints."
(K.G. Murty, for the American Mathematical Society, MathSciNet Mathematical Reviews on the web)

Spis treści

Preface. Part I: Preliminaries. 1. The linear programming problem. 2. The simplex method. 3. Large-scale LP problems. Part II: Computational Techniques. 4. Design principles of LP systems. 5. Data structures and basic operations. 6. Problem definition. 7. LP Processing. 8. Basis inverse, factorization. 9. The primal algorithm. 10. The dual algorithm. 11. Various issues. Index.