Fundamentals of Convex Analysis: Duality, Separation, Representation, and Resolution

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 296 pages
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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Contents

PRELIMINARY MATHEMATICS
1
CONVEX SETS IN R
35
SEPARATION AND SUPPORT THEOREMS
49
CONVEX CONES IN R
77
EXISTENCE THEOREMS FOR LINEAR SYSTEMS
121
THEOREMS OF
133
BASIC SOLUTIONS AND COMPLEMENTARY
165
EXTREME POINTS
189
SIMPLICIAL TOPOLOGY
235
REFERENCES
281
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