Linear and Nonlinear Programming

This new edition covers the central concepts of practical optimization techniques, with an emphasis on methods that are both state-of-the-art and popular. One major insight is the connection between the purely analytical character of an optimization problem and the behavior of algorithms used to solve a problem. This was a major theme of the first edition of this book and the fourth edition expands and further illustrates this relationship. As in the earlier editions, the material in this fourth edition is organized into three separate parts. Part I is a self-contained introduction to linear programming. The presentation in this part is fairly conventional, covering the main elements of the underlying theory of linear programming, many of the most effective numerical algorithms, and many of its important special applications. Part II, which is independent of Part I, covers the theory of unconstrained optimization, including both derivations of the appropriate optimality conditions and an introduction to basic algorithms. This part of the book explores the general properties of algorithms and defines various notions of convergence. Part III extends the concepts developed in the second part to constrained optimization problems. Except for a few isolated sections, this part is also independent of Part I. It is possible to go directly into Parts II and III omitting Part I, and, in fact, the book has been used in this way in many universities.

New to this edition is a chapter devoted to Conic Linear Programming, a powerful generalization of Linear Programming. Indeed, many conic structures are possible and useful in a variety of applications. It must be recognized, however, that conic linear programming is an advanced topic, requiring special study.   Another important topic is an accelerated steepest descent method that exhibits superior convergence properties, and for this reason, has become quite popular. The proof of the convergence property for both standard and accelerated steepest descent methods are presented in Chapter 8.  As in previous editions, end-of-chapter exercises appear for all chapters.

From the reviews of the Third Edition:

“… this very well-written book is a classic textbook in Optimization. It should be present in the bookcase of each student, researcher, and specialist from the host of disciplines from which practical optimization applications are drawn.” (Jean-Jacques Strodiot, Zentralblatt MATH, Vol. 1207, 2011)

Table of Contents

Chapter 1 Introduction

Part I Linear Programming
Chapter 2 Basic Properties of Linear Programs
Chapter 3 The Simplex Method
Chapter 4 Duality and Complementarity
Chapter 5 Interior-Point Methods
Chapter 6 Conic Linear Programming

Part II Unconstrained Problems
Chapter 7 Basic Properties of Solutions and Algorithms
Chapter 8 Basic Descent Methods
Chapter 9 Conjugate Direction Methods
Chapter 10 Quasi-Newton Methods

Part III Constrained Minimization
Chapter 11 Constrained Minimization Conditions
Chapter 12 Primal Methods
Chapter 13 Penalty and Barrier Methods
Chapter 14 Duality and Dual Methods
Chapter 15 Primal-Dual Methods

Appendix A Mathematical Review
Appendix B Convex Sets
Appendix C Gaussian Elimination
Appendix D Basic Network Concepts