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A First Course in Optimization Theory, by Rangarajan K. Sundaram
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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.
- Sales Rank: #268777 in Books
- Brand: Brand: Cambridge University Press
- Published on: 1996-06-13
- Released on: 1996-08-29
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .79" w x 5.98" l, 1.50 pounds
- Binding: Paperback
- 376 pages
- Used Book in Good Condition
Review
'... the book is an excellent reference for self-studies, especially for students in business and economics.' H. Noltemeier, W�rzberg
From the Back Cover
A First Course in Optimization Theory 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 R(superscript n), 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. Each chapter contains a number of detailed examples explaining both the theory and its applications for first-year master's and graduate students. "Cookbook" procedures are accompanied by a discussion of when such methods are guaranteed to be successful, and equally importantly, when they could fail. Each result in the main body of the text is also accompanied by a complete proof. A preliminary chapter and three appendices are designed to keep the book mathematically self-contained.
Most helpful customer reviews
0 of 0 people found the following review helpful.
Not Self-Contained
By Nat
This book mostly has clearly written proofs and easy-to-follow explanations for students who have some experience in proofs or basic analysis. In my opinion, it is definitely not a book for someone who has only seen calculus.
However, I have three very large problems with this book.
The first, and most important, is that the book is not self contained. In many theorems in Chapter 1, the reader is asked to see Baby Rudin for the proof. While it's pretty easy to find a PDF of Baby Rudin online for free, this is still not ideal. First, because Rudin and this book use different terminology/symbols for the same concepts, so there is a bit of unnecessary complexity in figuring how out Rudin's proof fits into this book's theorem. For example, the proof of Theorem 1.21 in this book is left to the proof of Rudin's Theorem 2.41. However, Rudin relies on the concept of k-cells, which this book never speaks of. Second, and most importantly, Rudin's proofs rely on concepts that this book has not defined. For example, Theorem 1.28 in this book relies on the proof of Theorem 2.36 in Baby Rudin. However, the proof in Baby Rudin relies on the concept of an "open cover," which this book does not define until a couple theorems later!
The second complaint I have about this book is that sometimes the proofs are sloppy. For example, at the beginning of section 1.2.8, the author states that unbounded sets must be compact and states that an unbounded sequence cannot contain a convergent subsequence. Instead of explaining why, though, he simply puts "(why?)" in the text. While this might be great for a student who is following this book in a class or with an instructor, this is incredibly frustrating for someone who is studying on their own.
Finally, as far as I can tell, this book is missing any discussion of the Bolzano Weierstrass Theorem, which is an important result in economics.
Update: The definition of a determinant is wrong in this book. It states you should take the negative of even permutations when it should be of odd permutations.
36 of 39 people found the following review helpful.
Good introduction to the field of optimization
By Dr. Lee D. Carlson
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.
10 of 12 people found the following review helpful.
The title says it all
By A Customer
A first course in Optimization theory - that is what the book is. The target audience is those who are inetersted in the theory of optimization. Some familiarity with Mathematical Analysis and Matrix Algebra would be helpful; however the first chapter lays the mathematical foundation and a careful reading would enable the reader to tackle the rest of the book.
Previous reviews have made a chapter by chapter analysis of the book and hence I will just highlight some of the things I liked about the approach used by the author. Whenever a theorem is stated different examples are given to emphasize the points. For example when stating the Lagrange Theorem and Kuhn-Tucker theorem the author points out when the theorems fail and gives detailed examples to illustrate the ideas. The author often draws from examples in finance to illustrate the practical importance of the theory. The one I liked most was how a cost minimization problem was solved by reducing the solution space to a compact space and then applying the Weierstrass theorem. The author also shows how some of the "cookbook" procedures really work and warns the readers against potential pitfalls in applying such procedures. If you are planning to study optimization theory and are looking for a good entry point into the subject this book is for you.
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