| Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems (Classics in Applied Mathematics, 37) |  | Author: Joel N. Franklin
Publisher: Society for Industrial Mathematics
Category: Book
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Sales Rank: 1240249
Media: Paperback
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Pages: 297
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Dimensions (in): 8.9 x 5.9 x 0.6
ISBN: 0898715091
Dewey Decimal Number: 519.72
EAN: 9780898715095
ASIN: 0898715091
Publication Date: January 15, 2002
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| Editorial Reviews:
Product Description
Many advances have taken place in the field of combinatorial algorithms since Methods of Mathematical Economics first appeared two decades ago. Despite these advances and the development of new computing methods, several basic theories and methods remain important today for understanding mathematical programming and fixed-point theorems. In this easy-to-read classic, readers learn Wolfe's method, which remains useful for quadratic programming, and the Kuhn-Tucker theory, which underlies quadratic programming and most other nonlinear programming methods. In addition, the author presents multiobjective linear programming, which is being applied in environmental engineering and the social sciences. PThe book presents many useful applications to other branches of mathematics and to economics, and it contains many exercises and examples. The advanced mathematical results are proved clearly and completely. By providing the necessary proofs and presenting the material in a conversational style, Franklin made Methods of Mathematical Economics extremely popular among students. The addition of a list of errata, new to this edition, should add to the book's popularity as well as its usefulness both in the classroom and for individual study. PThe book has three chapters: "Linear Programming," "Nonlinear Programming," and "Fixed-Point Theorems." The first and third chapters include the economic equilibrium theorems of von Neumann and of J. F. Nash, while the second chapter includes Kuhn-Tucker theory and Wolfe's simplex algorithm for quadratic programming. The book concludes with easy, elementary proofs of the famous theorems of Brouwer, of Kakutani, and of Schauder. These fundamental results are usually proved only in advanced texts in topology, economic theory, and nonlinear analysis. PbAudience/bbr This book is intended for undergraduate and graduate students of mathematics and economics; it requires no background in these areas except an understanding of elementary calculus and linear algebra. PbContents /bbr Preface to the Classics Edition; Preface; Errata; Chapter 1: Linear Programming. Introduction to Linear Programming; Linear Programs and Their Duals; How the Dual Indicates Optimality; Basic Solutions; The Idea of the Simplex Methods; Separating Planes for Convex Sets; Finite Cones and the Farkas Alternative; The Duality Principle; Perturbations and Parametric Programming; The Simplex Tableau Algorithm; The Revised Simplex Algorithm; A Simplex Algorithm for Degenerate Problems; Multiobjective Linear Programming; Zero-Sum, Two-Person Games; Integer Programming: Gomory's Method; Network Flows; Assignment and Shortest-Route Problems; The Transportation Problem; Chapter 2: Nonlinear Programming. Wolfe's Method for Quadratic Programming; Kuhn-Tucker Theory; Geometric Programming; Chapter 3: Fixed-Point Theorems. Introduction to Fixed Points; Contraction Mappings; Garsia's Proof of the Brouwer Fixed-Point Theorem; Milnor's Proof of the Brouwer Fixed-Point Theorem; Barycentric Coordinates, Sperner's Lemma, and an Elementary Proof of the Brouwer Fixed-Point Theorem; The Schauder Fixed-Point Theorem; Kakutani's Fixed-Point Theorem and Nash's Theorem for n-Person Games; Index.
Book Description
n this easy-to-read classic, readers learn Wolfe's method, which remains useful for quadratic programming, and the Kuhn-Tucker theory, which underlies quadratic programming and most other nonlinear programming methods. In addition, the author presents multiobjective linear programming, which is being applied in environmental engineering and the social sciences.
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| Customer Reviews:
Outdated, but has a unique collection of interesting topics July 25, 2008
This text attempts to survey the core subjects in optimization and mathematical economics: linear and nonlinear programming, separating plane theorems, fixed-point theorems, and some of their applications.br /br /This text covers only two subjects well: linear programming and fixed-point theorems. The sections on linear programming are centered around deriving methods based on the simplex algorithm as well as some of the standard LP problems, such as network flows and transportation problem. I never had time to read the section on the fixed-point theorems, but I think it could prove to be useful to research economists who work in microeconomic theory. This section presents four different proofs of Brouwer fixed-point theorem, a proof of Kakutani's Fixed-Point Theorem, and concludes with a proof of Nash's Theorem for n-person Games. br / br /Unfortunately, the most important math tools in use by economists today, nonlinear programming and comparative statics, are barely mentioned. This text has exactly one 15-page chapter on nonlinear programming. This chapter derives the Kuhn-Tucker conditions but says nothing about the second order conditions or comparative statics results.br /br /Most likely, the strange selection and coverage of topics (linear programming takes more than half of the text) simply reflects the fact that the original edition came out in 1980 and also that the author is really an applied mathematician, not an economist. This text is worth a look if you would like to understand fixed-point theorems or how the simplex algorithm works and its applications. Look elsewhere for nonlinear programming or more recent developments in linear programming.
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