Franklin J.N. Methods of mathematical economics: Linear and Nonlinear Programming, Fixed-Point Theorems

Society for Industrial and Applied Mathematics – 2002, 212 pages
ISBN: 0898715091, 9780898715095Many 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.
The book presents many useful applications…
The 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.AudienceThis 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.Preface to the Classics EditionErrata
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
Nonlinear Programming. Wolfe's Method for Quadratic Programming
Kuhn-Tucker Theory
Geometric Programming
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