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9781402006883

Complementarity, Equilibrium, Efficiency and Economics

by ; ;
  • ISBN13:

    9781402006883

  • ISBN10:

    1402006888

  • Format: Hardcover
  • Copyright: 2002-08-01
  • Publisher: Kluwer Academic Pub
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Summary

In complementarity theory, which is a relatively new domain of applied mathematics, several kinds of mathematical models and problems related to the study of equilibrium are considered from the point of view of physics as well as economics. In this book the authors have combined complementarity theory, equilibrium of economical systems, and efficiency in Pareto's sense. The authors discuss the use of complementarity theory in the study of equilibrium of economic systems and present results they have obtained. In addition the authors present several new results in complementarity theory and several numerical methods for solving complementarity problems associated with the study of economic equilibrium. The most important notions of Pareto efficiency are also presented. Audience: Researchers and graduate students interested in complementarity theory, in economics, in optimization, and in applied mathematics.

Table of Contents

Introduction
3(16)
Multi-Valued Mappings and Variational Inequalities
3(4)
Variational Inequalities on Convex Compact Sets
7(4)
Noncompact Case
11(5)
Comments
16(3)
Optimization Models
19(24)
Minimization Problem: General Form
19(7)
Convex Programming Problem
26(4)
Marginal Values and Feedback in Optimization Problems
30(5)
Multi-Objective Problem
35(5)
Comments
40(3)
General Economic Equilibrium
43(16)
Elements of Games Theory
43(3)
Model of Decentralized Economy, and Perfect Competition Equilibrium
46(5)
Arrow-Debreu Model
51(6)
Comments
57(2)
Models of Oligopoly
59(36)
Introduction
60(1)
Extended Cournot Models
60(15)
Problem Specification
60(5)
Existence Theorem
65(3)
Uniqueness of Solution
68(2)
Case of Constant Elasticity
70(3)
Influence Quotient as Function of Total Bargain Volume
73(2)
Network Models of Oligopoly
75(20)
Introduction
75(1)
Model with Individual Markets of Production Factors
76(3)
Existence and Uniqueness Theorems
79(6)
Model with a Common Market of Production Factors
85(3)
Human Migration Model
88(4)
Concluding Remarks
92(3)
Oligopoly with Leaders
95(18)
Market with Several Leaders
95(3)
Comparison of Cournot and Stackelberg Models
98(4)
Examples of Models
102(6)
Cournot Model Versus High Expectations Model
102(2)
Cournot Model Versus Mixed Conjectures Model
104(1)
Comparison of Stackelberg and Cournot Models
105(1)
Cournot Oligopoly Versus Perfect Competition
106(2)
Computing the Stackelberg Equilibrium
108(1)
Comments
109(4)
Complementarity Problems with Respect to General Cones
113(36)
Introduction and Preliminaries
113(3)
Complementarity Problem with Respect to a General Cone
116(3)
Sufficient Existence Conditions
119(7)
Standard Complementarity Problem
126(3)
Implicit Complementarity Problem
129(5)
General Order Complementarity Problem
134(1)
Semidefinite Complementarity Problem
135(14)
Problem Specification
135(1)
Theorems of Alternatives
136(5)
Existence of Solutions
141(6)
Conclusion
147(2)
Pseudomonotone and Implicit Complementarity Problems
149(48)
Introduction and Preliminaries
149(1)
Introduction and Preliminaries
150(3)
Preliminaries
151(1)
Leray-Schauder Type Alternative
152(1)
Complementarity Problems with Pseudomonotone Mappings
153(7)
Strict Feasibility, Solvability, and Exceptional Families of Elements
156(4)
Solving Bilevel Variational Inequalities with Monotone Mappings
160(5)
Problem Specification
160(1)
Existence Theorem
161(2)
Penalization Approach
163(2)
A Regularization Approach to Variational Inequalities with Pseudomonotone Mappings
165(8)
Preliminaries
166(1)
Regularization
167(6)
Concluding Remarks
173(1)
Infinite Dimensional Implicit Complementarity Problem
173(8)
Existence Alternatives
174(2)
Pairs of Mappings without Exceptional Families of Elements
176(2)
Continuous Selections Approach to Multi-Valued Implicit Complementarity Problem
178(3)
Compactification Approach to Micp with Upper Semicontinuous Mappings
181(16)
Complementarity Pivot Methods
197(34)
Introduction
197(1)
Lemke's Pivoting Algorithm
198(8)
Notation
198(1)
Lexico-Positive Inequality Systems
199(1)
The Linear Complementarity Problems Q|M and d|Q|M
199(1)
Complementary Bases and Adjacency
200(1)
Degeneracy
200(1)
Description of Lemke's Algorithm
201(1)
Secondary Rays and the Basic Theorem
202(1)
The Class L
202(3)
The Class Po
205(1)
Extensions of Lemke's Algorithm
206(12)
Projective Transformations
207(1)
Algorithms
208(7)
Properties of the Algorithms
215(3)
Block-Pivot Methods
218(5)
Pigeonholing
218(3)
Permutations
221(1)
Discussion
222(1)
Summary: Conditions Under a which the Complementarity Pivot Algorithm Works
223(8)
Results on LCPs Associated with Copositive Plus Matrices
224(1)
Results On LCPs Associates with L-and L-Matrices
225(1)
A Variant of the Complementarity Pivot Algorithm
226(5)
Scarf Type Algorithms
231(20)
Introduction
232(1)
Problem Specification
233(2)
Existence Conditions
235(2)
The Scarf Type Algorithm
237(5)
Constructive Proofs of the Existence Theorems
242(2)
Fixed Point Theorems
244(3)
Comments
247(4)
Newton-Like Methods
251(22)
Problem Specification
251(3)
Individual Response Functions
254(2)
Description of Algorithm
256(2)
Convergence Rate
258(1)
Extension to the Case of Nonlinear Costs
259(5)
Description of the Extended Algorithm
261(3)
Optimal Control of Step Precision in Bilevel Iteration Procedures of Newton Type
264(9)
Parametrization and Reduction to Nonlinear Equations
273(26)
Existence of Solutions to Perturbed Problem
274(4)
Convergence Rate
278(2)
Solving Perturbed Problems
280(3)
Regularization Methods
283(7)
Preliminaries
285(2)
Existence of Regularized Solution
287(3)
Inexact Regularization Methods
290(9)
Final Remarks
295(4)
Efficiency
299(88)
Introduction
299(1)
Preliminaries: A Minimal Background on Convex Cones
300(16)
Some Nations of Efficiency
316(17)
Some Existence Theorems for Efficiency
333(7)
Nuclear Cones and Efficiency
340(8)
A General Constructive Existence Principle for Efficiency
348(6)
Some Topological Properties of Efficient Point Sets
354(20)
Efficient Points and the Choquet Boundary
374(2)
Nuclear Cones, Pareto Efficiency and a Geometrical Aspect of Ekeland's Principle
376(11)
Approximative Efficiency
387
ε-Efficient Points
387
H-Efficency
392
Infinitesimal Efficiency
400
ε-Efficiency with Respect to an Arbitrary Set
402
Ekeland Variational Principle Types for Vector Valued Mappings, Efficiency and Approximative Efficency
406
Approximative Efficiency by a Perturbation of Cone
423
Efficiency in Product Spaces and ε-Efficiency
425

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