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Fundamentals of Convex Analysis : Duality, Separation, Representation, and Resolution - Michael J. Panik

Fundamentals of Convex Analysis

Duality, Separation, Representation, and Resolution

Hardcover

Published: 30th June 1993
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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.

Preface
An Overview
A Note on the Method of Mathematical Induction
Vectors Notation
Acknowledgements
Preliminary Mathematicsp. 1
Vector Spaces and Subspacesp. 1
The Solution Set of a System of Simultaneous Linear Equationsp. 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 Setsp. 35
Convex Combinationp. 39
Convex Hullp. 41
Separation and Support Theoremsp. 49
Hyperplanes and Half-planes Revisitedp. 49
Existence of Separating and Supporting Hyperplanesp. 52
Separation Renders Disjoint Alternativesp. 68
Convex Cones in R[superscript n]p. 77
Convex Conesp. 77
Finite Conesp. 83
Conical Hullp. 94
Extreme Vectors, Half-lines, and Half-spacesp. 98
Extreme Solutions of Homogeneous Linear Inequalitiesp. 101
Sum Cone and Intersection Cone Equivalencep. 105
Additional Duality Results for Finite Conesp. 110
Separation of Conesp. 114
Existence Theorems for Linear Systemsp. 121
Dual Homogeneous Linear Relationsp. 121
Existence Theoremsp. 122
Theorems of the Alternative for Linear Systemsp. 133
The Structure of a Theorem of the Alternativep. 133
Theorems of the Alternativep. 134
Homogeneous Inequalities/Equalities Under Convex Combinationp. 156
Basic Solutions and Complementary Slackness in Pairs of Dual Systemsp. 165
Basic Solutions to Linear Equalitiesp. 165
Moving From One Basic (Feasible) Solution to Anotherp. 172
Complementary Slackness in Pairs of Dual Systemsp. 184
Extreme Points and Directions for Convex Setsp. 189
Extreme Points and Directions for General Convex Setsp. 189
Convex Hulls Revisitedp. 196
Faces of Polyhedral Convex Sets: Extreme Points, Facets, and Edgesp. 199
Extreme Point Representation for Polyhedral Convex Setsp. 208
Directions for Polyhedral Convex Setsp. 214
Combined Extreme Point and Extreme Direction Representation for Polyhedral Convex Setsp. 220
Resolution of Convex Polyhedrap. 224
Separation of Convex Polyhedrap. 229
Simplicial Topology and Fixed Point Theoremsp. 235
Simplexesp. 235
Simplicial Decomposition and Subdivisionp. 239
Simplicial Mappings and Labelingp. 246
The Existence of Fixed Pointsp. 251
Fixed Points of Compact Point-to-Point Functionsp. 258
Fixed Points of Point-to-Set Functionsp. 261
Appendix: Continuous and Hemicontinuous Functionsp. 267
Referencesp. 281
Notation Indexp. 285
Indexp. 289
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792322795
ISBN-10: 0792322797
Series: THEORY AND DECISION LIBRARY SERIES B, MATHEMATICAL AND STATISTICAL METHODS
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 296
Published: 30th June 1993
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 25.4 x 17.15  x 2.54
Weight (kg): 0.7