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Linear programming: Foundations and Extensions (5th ed.)
Vanderbei R., Springer International Publishing, New York, NY, 2020. 495 pp. Type: Book (978-3-030394-14-1)
Date Reviewed: Mar 16 2021

Linear programming (LP) finds the optimal solution to a mathematical problem with given resources or constraints. The optimization of resources in achieving an optimal solution employs various methods; simplex, convex analysis, degeneracy, game theory, and many others are covered here. This book derives a reduced Karush-Kuhn-Tucker (KKT) system and extends the implementation of LP to data science and financial applications.

The book’s 25 chapters cover fundamental methods, with an introduction in chapter 1. Chapter 2 covers the simplex method, with various examples to illustrate its implementation. Chapter 3 discusses degeneracy, cycling, and pivoting rules. In addition, the author calculates the efficiency of the simplex method in finding an optimal solution to any LP problem in chapter 4. Chapter 5 is on the theoretical aspects of duality theory. The representation of primal and dual problems in matrix notation is emphasized in chapter 6. Further, chapter 7 explores the parametric self-dual simplex method with an introduction to sensitivity analysis.

Chapter 8 categorizes implementation issues, and chapter 9 highlights challenges to representing linear problems in the general form in the context of real-world problems. Furthermore, in chapter 10, the author discusses the dependency of fundamental results of convex analysis to LP. Chapter 11 illustrates the famous matrix game in proving the minimax theorem. Chapters 12 and 13 explore applications of LP to statistics, machine learning, optical character recognition, and quantitative finance.

In chapter 14, the author formulates a LP problem into a minimum-cost flow problem, which is then applied to various problems, including the transportation problem in chapter 15 and optimal structural design in chapter 16. Chapters 17 and 18 introduce a path-following method. Chapter 19 breaks a larger system into a reduced KKT system and normal equations, and chapter 20 covers its implementation issues. Chapter 21 explains the affine scaling principle for optimality and feasibility, and chapters 22 through 25 describe integer, quadratic, and convex programming problems and the homogeneous self-dual method.

This book is an interesting read for students, academics, and professionals who work to implement LP methods in order to achieve optimal solutions to their problems. It is for both novices and advanced professionals, with or without a mathematical background. The detailed examples make the reading worthwhile.

Reviewer:  Lalit Saxena Review #: CR147217 (2106-0134)
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