Computational Techniques of the Simplex Method

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: theoretical developments in the underlying algorithms; inclusion of results of computer science; using the principles of software engineering; and taking into account the state-of-the-art in computer technology. This book offers a systematic treatment focused on the computational issues of the simplex method. The purpose is to provide sufficient details about all aspects that are necessary for its successful implementation. On the basis of the book's systematic treatment, high quality implementations will result. Additionally, many of the procedures and techniques presented here can also be used in the efficient implementation of other optimization algorithms.