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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'... the book is an excellent reference for self-studies, especially for students in business and economics.' H. Noltemeier, Würzberg
Book Description
Divided into three separate parts, this book introduces students to optimization theory and its use in economics and allied disciplines. A preliminary chapter and three appendices are designed to keep the book mathematically self-contained.
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29 of 31 people found the following review helpful:
4.0 out of 5 stars
Good introduction to the field of optimization,
By Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews (VINE VOICE) (HALL OF FAME REVIEWER) (REAL NAME)
This review is from: A First Course in Optimization Theory (Paperback)
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.
28 of 31 people found the following review helpful:
5.0 out of 5 stars
Great book and an even greater value,
This review is from: A First Course in Optimization Theory (Paperback)
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,
By A Customer
This review is from: A First Course in Optimization Theory (Paperback)
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.
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Inside This Book
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First Sentence:
This chapter lays the mathematical foundation for the study of optimization that occupies the rest of this book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
parametric monotonicity, satisfies strictly increasing differences, compact sublattice, unconstrained local maximum, convex maximization problem, constraint qualification, convex graph, expenditure minimization problem, strict local maximum, cookbook procedure, convexity restrictions, parametric continuity, unique global maximum, stationary strategy, technical digression, dynamic programming problem, notational ease, unique critical point, utility maximization problem, lim supk, subsidiary hypotheses, lim infk, arbitrary index set, global optima, strict local minimum Key Phrases - Capitalized Phrases (CAPs): (learn more)
Weierstrass Theorem, Maximum Theorem, Theorem of Lagrange, Bellman Equation, Proof Let, Proof See Section, Proof Suppose, Proof See Rudin, Intermediate Value Theorem, Implicit Function Theorem, Mean Value Theorem, Taylor's Theorem, Tarski's Fixed Point Theorem, Contraction Mapping Lemma, Proof First, Proof of the Kuhn-Tucker Theorem, Proof See Munkres, Bellman Principle of Optimality, Proof Fix, Proof Pick, Proof See Debreu, Proof See Johnston New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
This chapter lays the mathematical foundation for the study of optimization that occupies the rest of this book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
parametric monotonicity, satisfies strictly increasing differences, compact sublattice, unconstrained local maximum, convex maximization problem, constraint qualification, convex graph, expenditure minimization problem, strict local maximum, cookbook procedure, convexity restrictions, parametric continuity, unique global maximum, stationary strategy, technical digression, dynamic programming problem, notational ease, unique critical point, utility maximization problem, lim supk, subsidiary hypotheses, lim infk, arbitrary index set, global optima, strict local minimum Key Phrases - Capitalized Phrases (CAPs): (learn more)
Weierstrass Theorem, Maximum Theorem, Theorem of Lagrange, Bellman Equation, Proof Let, Proof See Section, Proof Suppose, Proof See Rudin, Intermediate Value Theorem, Implicit Function Theorem, Mean Value Theorem, Taylor's Theorem, Tarski's Fixed Point Theorem, Contraction Mapping Lemma, Proof First, Proof of the Kuhn-Tucker Theorem, Proof See Munkres, Bellman Principle of Optimality, Proof Fix, Proof Pick, Proof See Debreu, Proof See Johnston New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 15
books:
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11
books
cite this book:
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- Dynamic Programming (Dover Books on Computer Science) by Richard Bellman on page 276, and page 281
- in Back Matter
- Probability and Measure, 3rd Edition by Patrick Billingsley in Back Matter
- Game Theory by Drew Fudenberg in Back Matter
- The Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics) by Walter Rudin in Back Matter
See all 15 books this book cites
- Capacity Reservation for Capital-intensive Technologies: An Options Approach (Lecture Notes in Economics and Mathematical Systems) by Stefan Spinler in Back Matter
- Quantitative Models for Supply Chain Management (International Series in Operations Research & Management Science) by Sridhar Tayur on page 551
- Bargaining Theory with Applications by Abhinay Muthoo in Back Matter
- Microfoundations of Financial Economics: An Introduction to General Equilibrium Asset Pricing (Princeton Series in Finance) by Yvan Lengwiler in Back Matter
- The Theory and Practice of Revenue Management (International Series in Operations Research & Management Science) by Kalyan T. Talluri in Back Matter
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