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Introduction to

Linear Optimization

by Dimitris Bertsimas and John Tsitsiklis

Athena Scientific & Dynamic Ideas,
Belmont, Massachusetts, 1997.

ISBN: 978-1-886529-19-9

The purpose of this book is to provide a unified, insightful, and modern treatment of linear optimization, that is, linear programming, network flow problems, and discrete linear optimization. We discuss both classical topics, as well as the state of the art. We give special attention to theory, but also cover applications and present case studies. Our main objective is to help the reader become a sophisticated practitioner of (linear) optimization, or a researcher.


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Educational Philosophy Of the Book

Our general philosophy is that insight matters most. For the subject matter of this book, this necessarily requires a geometric view. On the other hand, problems are solved by algorithms, and these can only be described algebraically. Hence, our focus is on the beautiful interplay between algebra and geometry. We build understanding using figures and geometric arguments, and then translate ideas into algebraic formulas and algorithms. Given enough time, we expect that the reader will develop the ability to pass from one domain to the other without much effort.

Another of our objectives is to be comprehensive, but economical. We have made an effort to cover and highlight all of the principal ideas in this field. However, we have not tried to be encyclopedic, or to discuss every possible detail relevant to a particular algorithm. Our premise is that once mature understanding of the basic principles is in place, further details can be acquired by the reader with little additional effort.

Our last objective is to bring the reader up to date with respect to the state of the art. This is especially true in our treatment of interior point methods, large scale optimization, and the presentation of case studies that stretch the limits of currently available algorithms and computers.

Distinguishing Characteristics Of This Book

The success of any optimization methodology hinges on its ability to deal with large and important problems. In that sense, the last chapter, on the art of linear optimization, is a critical part of this book.

It will, we hope, convince the reader that progress on challenging problems requires both problem specific insight, as well as a deeper understanding of the underlying theory.

In any book dealing with linear programming, there are some important choices to be made regarding the treatment of the simplex method. Traditionally, the simplex method is developed in terms of the full simplex tableau, which tends to become the central topic. We have found that the full simplex tableau is a useful device for working out numerical examples. But other than that, we have tried not to overemphasize its importance.

Let us also mention another departure from many other textbooks. Introductory treatments often focus on standard form problems, which is sufficient for the purposes of the simplex method. On the other hand, this approach often leaves the reader wondering whether certain properties are generally true, and can hinder the deeper understanding of the subject. We depart from this tradition: we consider the general form of linear programming problems and define key concepts (e.g., extreme points) within this context. (Of course, when it comes to algorithms, we often have to specialize to the standard form.) In the same spirit, we separate the structural understanding of linear programming from the particulars of the simplex method. For example, we include a derivation of duality theory that does not rely on the simplex method.

Finally, this book contains a treatment of several important topics that are not commonly covered. These include a discussion of the column geometry and of the insights it provides into the efficiency of the simplex method, the connection between duality and the pricing of financial assets, a unified view of delayed column generation and cutting plane methods, stochastic programming and Benders decomposition, the auction algorithm for the assignment problem, certain theoretical implications of the ellipsoid algorithm, a thorough treatment of interior point methods, and a whole chapter on the practice of linear optimization. There are also several noteworthy topics that are covered in the exercises, such as Leontief systems, strict complementarity, options pricing, von Neumann's algorithm, submodular function minimization, and bounds for a number of integer programming problems.

Chapter Descriptions

An important theme that runs through several chapters is the modeling, complexity, and algorithms for problems with an exponential number constraints. We discuss modeling in Section~10.3, complexity in Section~8.5, algorithmic approaches in Chapter~6 and~8.5, and we conclude with a case study in Section~12.5.

There are a fair number of exercises that are given at the end of each chapter. Most of them are intended to deepen the understanding of the subject, or to explore extensions of the theory in the text, as opposed to routine drills. However, several numerical exercises are also included. Starred exercises are supposed to be fairly hard. A solutions manual for qualified instructors can be obtained from the authors.

Chapter 1: Introduces the linear programming problem, together with a number of examples, and provides some background material on linear algebra.

Chapter 2: Deals with the basic geometric properties of polyhedra, focusing on the definition and the existence of extreme points, and emphasizing the interplay between the geometric and the algebraic viewpoints.

Chapter 3: Contains more or less the classical material associated with the simplex method, as well as a discussion of the column geometry. It starts with a high-level and geometrically motivated derivation of the simplex method. It then introduces the revised simplex method, and concludes with the simplex tableau. The usual topics of Phase I and anticycling are also covered.

Chapter 4: It is a comprehensive treatment of linear programming duality. The duality theorem is first obtained as a corollary of the simplex method. A more abstract derivation is also provided, based on the separating hyperplane theorem, which is developed from first principles. It ends with a deeper look into the geometry of polyhedra.

Chapter 5: Discusses sensitivity analysis, that is, the dependence of solutions and the optimal cost on the problem data, including parametric programming. It also develops a characterization of dual optimal solutions as subgradients of a suitably defined optimal cost function.

Chapter 6: Presents the complementary ideas of delayed column generation and cutting planes. These methods are first developed at a high level, and are then made concrete by discussing the cutting stock problem, Dantzig-Wolfe decomposition, stochastic programming, and Benders decomposition.

Chapter 7: Provides a comprehensive review of the principal results and methods for the different variants of the network flow problem. It contains representatives from all major types of algorithms: primal descent (the simplex method), dual ascent (the primal-dual method), and approximate dual ascent (the auction algorithm). The focus is on the major algorithmic ideas, rather than on the refinements that can lead to better complexity estimates.

Chapter 8: Includes a discussion of complexity, a development of the ellipsoid method, and a proof of the polynomiality of linear programming. It also discusses the equivalence of separation and optimization, and provides examples where the ellipsoid algorithm can be used to derive polynomial time results for problems involving an exponential number of constraints.

Chapter 9: Contains an overview of all major classes of interior point methods, including affine scaling, potential reduction, and path following (both primal and primal-dual) methods. It includes a discussion of the underlying geometric ideas and computational issues, as well as convergence proofs and complexity analysis.

Chapter 10: Introduces integer programming formulations of discrete optimization problems. It provides a number of examples, as well as some intuition as to what constitutes a a “strong” formulation.

Chapter 11: Covers the major classes of integer programming algorithms, including exact methods (branch and bound, cutting planes, dynamic programming), approximation algorithms, and heuristic methods (local search and simulated annealing). It also introduces a duality theory for integer programming.

Chapter 12: Deals with the art in linear optimization, i.e., the process of modeling, exploiting problem structure, and fine tuning of optimization algorithms. We discuss the relative performance of interior point methods and different variants of the simplex method, in a realistic large scale setting. We also give some indication of the size of problems that can be currently solved.

About The Authors

John N. Tsitsiklis is a Clarence J Lebel Professor of Electrical Engineering, with the Department of Electrical Engineering and Computer Science (EECS) at MIT. He is affiliated with the Laboratory for Information and Decision Systems (LIDS) and the Operations Research Center (ORC).

Dimitris Bertsimas is the Boeing Professor of Operations Research at the Massachusetts Institute of Technology and Codirector of the MIT Operations Research Center.