Preface

This book is about optimization problems that arise in the field of operations research, including linear optimization (continuous and discrete) and convex programming. Linear programming plays a central role because these problems can be solved very efficiently; it also has useful connections with discrete and convex optimization. Convex optimization is not included in many books at this level. However, in the past three decades new algorithms and many new applications have increased interest in convex optimization. Like linear programming, large applied problems can now be solved efficiently and reliably. Conceptually, convex programming fits better with linear programming than with general nonlinear programming.

These types of optimization are also appropriate for this book because they have a clear theory and unifying mathematical principles, much of which is included. The approach taken has three emphases.

1. Modeling is covered through a broad range of applications to fields such as on-line marketing and inventory management, retail pricing, humanitarian response and rural development, public sector planning, health delivery, finance, manufacturing, service systems, and transportation. Many of these tell the story of successful applications of operations research.
2. A mathematical approach is used to communicate in a concise, unified manner. Matrix notation is introduced early and used extensively. Questions of correctness are not glossed over; the mathematical issues are described and, where the level is appropriate, proofs presented. Connections are made with some other topics in undergraduate mathematics. This approach grew out of my 30 years of teaching these topics to undergraduate mathematics students.
3. The reasoning behind algorithms is presented. Rather than introducing algorithms as arbitrary procedures, whenever possible reasons are given for why one might design such an algorithm. Enough analysis of algorithms is presented to give a basic understanding of complexity of algorithms and what makes an algorithm efficient. Algorithmic thinking is taught, not assumed.

Many introductory operations research textbooks emphasize models and algorithms without justifications and without making use of mathematical language; such books are ill-suited for mathematics students. On the other hand, texts that treat the subject more mathematically tend to be too advanced and detailed, at the expense of applications. This book seeks a middle ground.

The intended audience is junior and senior mathematics students in a course on optimization or (deterministic) operations research. The background required is a good knowledge of linear algebra and, in a few places, some calculus. These are reviewed in the appendix. The coverage and approach is intentionally kept at an undergraduate level. Material is often organized by depth, so that more advanced topics or approaches appear at the end of sections and chapters and are not needed for continuity. For example, the many ways to speed up the simplex method are saved for the last section of Chapter 9.

In keeping with this audience, the number of different problem types and algorithms is kept to a minimum, emphasizing instead unified approaches and more general problems. In particular, heuristic algorithms are only mentioned briefly. They are used for hard problems and use many different approaches, while this book focuses on problems that have efficient algorithms or at least unified approaches. The goal is to introduce students to optimization, not to be a thorough reference, and to appeal to students who are curious about other uses of mathematics. The many applications in the early chapters make the case that optimization is useful. The latter chapters connect the solution of these problems to the linear algebra and other mathematics that this audience is familiar with.

The book is also specifically written for the instructor who is mathematically trained, not for a specialist in operations research and optimization. The careful treatment of algorithmic thinking and the introduction to complexity of algorithms are intended to assist these instructors. The mathematical style throughout the book should be more accommodating to mathematics professors. It is also intended to support learning objectives more likely to be found in a mathematics department, including why the algorithms are correct and how they use theoretical results such as convexity and duality. Being able to perform an algorithm by hand is not a primary objective; it plays a supporting role to understanding the notation and reasoning of the algorithm. Calculations that are well-understood by mathematics students, such as solving a linear system or performing row operations, are not elaborated on. The somewhat more advanced material at the end of sections or chapters is also intended to support instructors who are not specialists, allowing them to extend their knowledge and explore the literature.

Chapters 1-4 are devoted to introducing optimization and optimization modeling. Convex models appear later with the other material on convex optimization. In my experience teaching mathematics students, they find modeling challenging. These chapters assume technology is available to solve problems, so that the focus can stay on formulation, as well as interpreting solutions. They build steadily in sophistication, starting with numerical instances but soon moving to algebraic formulations to make clear the distinction between model structure and data. The features of the models also build, particularly when using logical variables. In contrast with the business case study approach, each model has a limited number of features and focuses on some novel feature. I have found that mathematics students relate better to a succession of simpler models, from which they learn different modeling principles, than to a long case study.

Chapters 5-8 discuss iterative algorithms, giving some easily explained examples from discrete optimization, and the theoretical background for linear programming. This includes a little computational complexity, convexity and the study of polyhedra, optimality conditions for linear programming, and duality theory for linear programming. It is unorthodox to cover all of these before introducing the simplex method. However, conceptually these topics fit together and do not depend on the simplex method; putting them together emphasizes this fact. Chapter 8 on duality is independent of Chapter 9 on the simplex method, so that they can be covered them in either order. I typically skip the topics in Sections 5.3, 7.5, 8.4, and 8.5 to arrive at the simplex method about the middle of the semester.

Chapters 9-11 present the simplex method, including sensitivity analysis, and other algorithms for linear programming. A developmental approach is taken to presenting the simplex method. Starting with geometric reasoning about why it works, it is presented in the "naive" form first, where the so-called inverse basis matrix is computed from scratch each iteration. While this form is computationally inefficient, it is very easy to explain to a mathematics student, both for computing and justifying that the algorithm works. When working examples, technology can be used to invert and multiply matrices. After completing the picture of why the method works (degeneracy, two-phase simplex), Section 9.4 takes up the issue of making the simplex method more efficient, including the tableau form and revised simplex method. Instructors who wish to start with the tableau can use the material found here. Chapter 10 on sensitivity analysis, which depends Chapter 8, can be skipped without loss of continuity; however, the interpretation of dual variables as shadow prices and partial derivatives is enriching, even in an age when sensitivity analysis can be done quickly by solving modified linear programs. An interpretation of strong duality in terms of average costs is also given in Section 10.3. Chapter 11 presents three more algorithms for linear programming, all of which rely on duality: the dual simplex, transportation simplex, and a primal-dual, or path following, interior point method. The transportation simplex method is presented first as a minimum cost flow problem, then specialized to transportation problems.

Chapters 12 and 13 present integer programming algorithms. These relate to the earlier material because of the importance of linear programming to establish bounds when solving an integer program. Integer programming also has greater modeling power, as demonstrated by the many applications in Chapter 14. Chapters 14 and 15 introduce convex programming, including some motivating applications. The earlier chapters are designed to prepare the reader to understand convex programming more readily. The KKT optimality conditions and duality theorems are a generalization of Lagrangian duality (Section 8.4). Necessary and sufficient conditions for a global optimum follow from convexity theory, already applied to linear programs in Chapter 6. Chapter 15 culminates in the primal-dual interior point method, which was presented for linear programs in Section 11.3. Quadratic programming is also introduced and the connection between the primal-dual interior point method and sequential quadratic programming is made.

Supplemental material will be available at the web site www.gordon.edu/michaelveatch/optimization for the book. A full solution...