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Introduction to the Theory of Optimization in Euclidean Space

Samia Challal
Chapman and Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Tia Sondjaja
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As the book’s back matter accurately describes, this book is “intended to provide students with a robust introduction to optimization in Euclidean space, demonstrating the theoretical aspects … whilst also providing clear proofs and applications”. More specifically, the textbook is suitable
for undergraduates and master students in mathematics and related fields who are interested in a more rigorous treatment of the mathematical foundation of optimization. Chapter 1 includes a short but self-contained overview of relevant bits of multivariate calculus, but the book might
better serve readers who have had some prior exposure to these topics.
The textbook is relatively compact and written in a lecture note style that would be suitable for either a course or an independent study. Each mathematical theorem and its proof are complemented by examples and exercises that are mostly concrete and computational. Even though the mathematical notation could be dense in some parts for beginning readers, the topic organization and exposition are clear throughout the book: each topic is well-motivated, and each section is accompanied by a good set of exercises and well-written solutions.
The book does a good job of incorporating applications, most of them from economics. Chapter 1 of the book includes several modeling examples that illustrate the importance of the main classes of optimization questions discussed in Chapters 2-5 (for example, the classical “diet problem” as an example of an inequality-constrained optimization question). However, readers who would like to focus specifically on the modeling, algorithmic, or applied aspects of optimization, would be better served by other books.
This book distinguishes itself among undergraduate optimization books by organically picking up where multivariable calculus leaves off, with regards to both topic selection and level of rigor and abstraction. While readers might have been exposed to unconstrained optimization (Chapter 2) and equality-constrained optimization (Chapter 3) by reading a standard multivariable calculus book, this book goes more in-depth into optimization, by providing proofs for each theorem and useful remarks. It also introduces readers to important ideas such as regular points, feasible directions, complementary slackness, and various necessary and sufficient conditions. The book culminates with inequality-constrained optimization in Chapter 4, which includes an introduction to the Karush-Kuhn-Tucker (KKT) conditions for a variety of inequality- and mixed-constrained optimization questions.
To my knowledge, there are few other textbooks like this. Most optimization books for undergraduates are written mostly with economics, operations research, or engineerinstudents in mind and put more emphasis on the modeling and algorithmic aspects of optimization (some of these books do include rigorous proofs as well of the relevant mathematical results); many books focus on only one class of optimization questions (for example, only on linear programming) while other books cover a wide breadth of topics that there is little space for a focused treatment of the mathematical foundations. There are great reference textbooks that include discussions of the mathematical underpinnings of optimization in euclidean spaces, but they are often too abstract and the organization of topics might not be intuitive to those new to optimization.

Tia Sondjaja ([email protected]) is a clinical faculty member in the math department at New York University. She did her graduate work in optimization and operations research.