Fundamentals of Convex Analysis

Duality, Separation, Representation, and Resolution

By: M.J. Panik

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Hardcover | 30 June 1993

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324 Pages

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Fundamentals of Convex Analysis offers an in-depth look at some of the fundamental themes covered within an area of mathematical analysis called convex analysis. In particular, it explores the topics of duality, separation, representation, and resolution. The work is intended for students of economics, management science, engineering, and mathematics who need exposure to the mathematical foundations of matrix games, optimization, and general equilibrium analysis. It is written at the advanced undergraduate to beginning graduate level and the only formal preparation required is some familiarity with set operations and with linear algebra and matrix theory. Fundamentals of Convex Analysis is self-contained in that a brief review of the essentials of these tool areas is provided in Chapter 1. Chapter exercises are also provided.
Topics covered include: convex sets and their properties; separation and support theorems; theorems of the alternative; convex cones; dual homogeneous systems; basic solutions and complementary slackness; extreme points and directions; resolution and representation of polyhedra; simplicial topology; and fixed point theorems, among others. A strength of this work is how these topics are developed in a fully integrated fashion.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Geometry Analytic Geometry Optimisation Game Theory Mathematical Foundation Set Theory Algebraic Geometry Linear Programming

Economics Economic Theory & Philosophy Applied Mathematics

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Published: 8th December 2010

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Preface
An Overview
A Note on the Method of Mathematical Induction
Vectors Notation
Acknowledgements
Preliminary Mathematics	p. 1
Vector Spaces and Subspaces	p. 1
The Solution Set of a System of Simultaneous Linear Equations	p. 8
Point-set Theory: Topological Properties of R[superscript n]	p. 13
Hyperplanes and Half-planes (-spaces)	p. 21
Convex Sets in R[superscript n]	p. 35
Convex Sets	p. 35
Convex Combination	p. 39
Convex Hull	p. 41
Separation and Support Theorems	p. 49
Hyperplanes and Half-planes Revisited	p. 49
Existence of Separating and Supporting Hyperplanes	p. 52
Separation Renders Disjoint Alternatives	p. 68
Convex Cones in R[superscript n]	p. 77
Convex Cones	p. 77
Finite Cones	p. 83
Conical Hull	p. 94
Extreme Vectors, Half-lines, and Half-spaces	p. 98
Extreme Solutions of Homogeneous Linear Inequalities	p. 101
Sum Cone and Intersection Cone Equivalence	p. 105
Additional Duality Results for Finite Cones	p. 110
Separation of Cones	p. 114
Existence Theorems for Linear Systems	p. 121
Dual Homogeneous Linear Relations	p. 121
Existence Theorems	p. 122
Theorems of the Alternative for Linear Systems	p. 133
The Structure of a Theorem of the Alternative	p. 133
Theorems of the Alternative	p. 134
Homogeneous Inequalities/Equalities Under Convex Combination	p. 156
Basic Solutions and Complementary Slackness in Pairs of Dual Systems	p. 165
Basic Solutions to Linear Equalities	p. 165
Moving From One Basic (Feasible) Solution to Another	p. 172
Complementary Slackness in Pairs of Dual Systems	p. 184
Extreme Points and Directions for Convex Sets	p. 189
Extreme Points and Directions for General Convex Sets	p. 189
Convex Hulls Revisited	p. 196
Faces of Polyhedral Convex Sets: Extreme Points, Facets, and Edges	p. 199
Extreme Point Representation for Polyhedral Convex Sets	p. 208
Directions for Polyhedral Convex Sets	p. 214
Combined Extreme Point and Extreme Direction Representation for Polyhedral Convex Sets	p. 220
Resolution of Convex Polyhedra	p. 224
Separation of Convex Polyhedra	p. 229
Simplicial Topology and Fixed Point Theorems	p. 235
Simplexes	p. 235
Simplicial Decomposition and Subdivision	p. 239
Simplicial Mappings and Labeling	p. 246
The Existence of Fixed Points	p. 251
Fixed Points of Compact Point-to-Point Functions	p. 258
Fixed Points of Point-to-Set Functions	p. 261
Appendix: Continuous and Hemicontinuous Functions	p. 267
References	p. 281
Notation Index	p. 285
Index	p. 289
Table of Contents provided by Blackwell. All Rights Reserved.

Fundamentals of Convex Analysis

Duality, Separation, Representation, and Resolution

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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Paperback

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