A First Course in Optimization Theory (Paperback)

by Rangarajan K. Sundaram (Author) "This chapter lays the mathematical foundation for the study of optimization that occupies the rest of this book..." (more)
Key Phrases: parametric monotonicity, satisfies strictly increasing differences, compact sublattice, Weierstrass Theorem, Maximum Theorem, Theorem of Lagrange (more...)

Editorial Reviews

Review
'... the book is an excellent reference for self-studies, especially for students in business and economics.' H. Noltemeier, Würzberg

Product Description
This book introduces students to optimization theory and its use in economics and allied disciplines. The first of its three parts examines the existence of solutions to optimization problems in Rn, and how these solutions may be identified. The second part explores how solutions to optimization problems change with changes in the underlying parameters, and the last part provides an extensive description of the fundamental principles of finite- and infinite-horizon dynamic programming. A preliminary chapter and three appendices are designed to keep the book mathematically self-contained.

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Product Details

• Paperback: 375 pages
• Publisher: Cambridge University Press (June 13, 1996)
• Language: English
• ISBN-10: 0521497701
• ISBN-13: 978-0521497701
• Product Dimensions: 9.9 x 6.8 x 0.9 inches
• Shipping Weight: 1.5 pounds (View shipping rates and policies)
• Average Customer Review: 4.6 out of 5 stars See all reviews (11 customer reviews)
• Amazon.com Sales Rank: #40,651 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

25 of 27 people found the following review helpful:

4.0 out of 5 stars Good introduction to the field of optimization, March 20, 2001

By	Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (TOP 100 REVIEWER)    (REAL NAME)

This book gives a nice introduction to the theory of optimization from a purely mathematical standpoint. The computational and algorithmic aspects of the subject are not treated, with emphasis instead placed on existencetheorems for various optimization problems. The author does an effective job of detailing the mathematical formalism needed in optimization theory. After a brief review of background mathematics in the first chapter, the author outlines the objectives of optimization theory in Chapter Two. He also gives some examples of optimization problems, such as utility maximization, expenditure minimization, profit maximization, cost minimization, and portfolio choice. All of these examples are extremely important in industrial, logistical, and financial applications. The author is also careful in this chapter to outline his intentions in later chapters, namely, that of finding the existence of solutions to optimization problems, and also in the characterization of the set of optimal points. The existence question is outlined in Chapter Three using only elementary calculus, and the Weierstrass theorem is proved. Necessary conditions for unconstrained optima are examined in the next chapter, again using only elementary calculus and linear algebra. Lagrange multipliers and how they are used in constrained optimization problems are effectively discussed in Chapter 5. To discuss how optimization problems vary with a set of parameters, in particular if they vary continuously with the set of parameters, the author introduces the concept of a corespondence. This is essentially a map that assigns sets to points. His discussion of upper and lower-semicontinuity is very clear and I think one of the best presentations given at this level. He then proves a maximum theorem, showing that parametrized optimization problems can have continuous solutions under certain conditions. A game-theoretic application follows along with statements, but not proofs, of the Kakutani and Brouwer Fixed Point theorems. The author introduces an order relation on the parameter space and discusses parametric monotonicity in the next chapter. Again a game theory application is given along with a statement (but not a proof) of the Tarski Fixed Point theorem. The last two chapters cover dynamic programming and these are the most interesting chapters of the book. It is here that the author makes the connection with more advanced treatments of optimization theory, via Banach spaces and nonlinear functional analysis. With further reading in real analysis and topology, readers will be well on their way to understanding more advanced treatments of optimization theory that use nonlinear functional analysis and differential topology.

27 of 30 people found the following review helpful:

5.0 out of 5 stars Great book and an even greater value, December 7, 1999

By	Andrew Cusano (Lincoln, NE USA) - See all my reviews

This book was organized and written with perfection. The explanations are remarkable and the "cookbook" procedures for Lagrange and K-T methods were great. I especially admired the fact that the author actually mentioned how these procedures could fail to yield an optimized value. This is worthwhile in today's university mathematics where one is simply taught to plug numbers into formulae and algorithms to get the desired answer. The book also slants towards optimization problems in economic theory as well as other disciplines. Finally, in an age when textbooks can easily run over $100, it was nice to see this book, filled with a wealth of information, so moderately priced.

15 of 17 people found the following review helpful:

5.0 out of 5 stars Excellent book for PhD students in Operations Management, September 2, 1999

By A Customer

This is an excellent book for anybody interested in non-linear optimization within economics framework. The book is self-contained and includes all the basic theory one needs to know to understand optimization. To my knowledge, this is the only book merging non-linear optimization with game theory and such concepts as supermodularity and parametric monotonicity.