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Linear programming using MATLAB
Ploskas N., Samaras N., Springer International Publishing, New York, NY, 2017. 637 pp. Type: Book (978-3-319659-17-6)
Date Reviewed: Nov 7 2018

Linear programming (LP) is an operational research method for finding the optimal values of a linear objective function under linear constraints. LP is recognized as one of the top ten algorithms of our time. LP problems extend not only to scientific fields, but also have direct practical applications in, for example, engineering (structural and mechanical engineering, electrical and power systems, transportation planning), computer science (clustering and classification, wireless and network systems), economics, and industry. This puts LP in the focus of current scientific and applied studies. One of the main advantages of the method is its ability to solve global optimality problems. Together with this, as a deterministic approach, with the increase in the dimensions of the problems and/or presence of singularities such as ill-conditionality, the calculation time, memory, and other computer resources required to find the optimal solution increase sharply. For this reason, various algorithms continue to be developed using the performance improvement methods for LP.

Linear programming using MATLAB is a typical and adequate example in this regard. The authors present the results on the current state and the most effective variants, combined with the appropriate calculation techniques for the simplex method, the main algorithm of LP, illustrated in detail using numerical examples and powerful MATLAB code.

The book is organized in 12 chapters and two appendices. Chapter 1 introduces basic LP concepts and notions. Chapter 2 presents the classical background of LP. A general form of the problem, its canonical form, and the equivalent duality problem are defined. Theoretical aspects are illustrated using suitable examples and MATLAB code. When n = 2, a graphical method is presented for finding the feasible area and optimal solution. Chapter 3 includes information about the software tools and formats used in the examples.

The following three chapters detail the basic procedures and analysis necessary to prepare the LP problem for the application of simplex algorithms. Chapter 4 examines presolve techniques, based on the mathematical basis of LP. Chapter 5 is devoted to 11 different scaling techniques used prior to the application of simplex algorithms. Chapter 6 describes the most common pivoting rules, which are a key step in the transition from one extreme point of the problem to another. The algorithm efficiency of all procedures has been assessed in advance (calculation time and iteration number) depending on the type of applied preprocessing, scaling, and pivoting rules and techniques, as well as by comparing the different algorithms.

Chapter 7 presents four basis inverse and update methods, including Gauss elimination and lower–upper (LU) decomposition for replacing one basic variable with a new one. This is all supported by numerical examples and the respective program implementations in MATLAB.

The next four chapters (8 through 11)--essentially the most important part of this book--present the revised primal and revised dual simplex algorithms and the exterior point and interior point simplex algorithms. Chapter 11 focuses on primal-dual interior point methods (IPMs), which are characterized by high computational performance. The most effective ones use Mehrotra’s predictor-corrector (MPC) method. The authors present the aforementioned algorithms with their mathematical background, numerical examples, and implementations in MATLAB. The final chapter (12) considers issues related to the sensitivity analysis of algorithms with regard to cost and right-hand side vectors and the coefficients of the constraint conditions.

The book’s style enables the reader to examine in detail the presented methods and calculation approaches. Its completeness and novelty could make it very useful for undergraduate students and junior researchers, as well as programmers and professionals in operational research.

Reviewer:  Snezhana Gocheva-Ilieva Review #: CR146310 (1902-0015)
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