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Full Description
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.
Contents
1. Preliminary Mathematics.- 1.1. Vector Spaces and Subspaces.- 1.2. The Solution Set of a System of Simultaneous Linear Equations.- 1.3. Point-set Theory: Topological Properties of Rn.- 1.4. Hyperplanes and Half-planes (-spaces).- 2. Convex Sets in Rn.- 2.1. Convex Sets.- 2.2. Convex Combination.- 2.3. Convex Hull.- 3. Separation and Support Theorems.- 3.1. Hyperplanes and Half-planes Revisited.- 3.2. Existence of Separating and Supporting Hyperplanes.- 3.3. Separation Renders Disjoint Alternatives.- 4. Convex Cones in Rn.- 4.1. Convex Cones.- 4.2. Finite Cones.- 4.3. Conical Hull.- 4.4. Extreme Vectors, Half-lines, and Half-spaces.- 4.5. Extreme Solutions of Homogeneous Linear Inequalities.- 4.6. Sum Cone and Intersection Cone Equivalence.- 4.7. Additional Duality Results for Finite Cones.- 4.8. Separation of Cones.- 5. Existence Theorems for Linear Systems.- 5.1. Dual Homogeneous Linear Relations.- 5.2. Existence Theorems.- 6. Theorems of the Alternative for Linear Systems.- 6.1. The Structure of a Theorem of the Alternative.- 6.2. Theorems of the Alternative.- 6.3. Homogeneous Inequalities/Equalities Under Convex Combination.- 7. Basic Solutions and Complementary Slackness in Pairs of Dual Systems.- 7.1. Basic Solutions to Linear Equalities.- 7.2. Moving From One Basic (Feasible) Solution to Another.- 7.3. Complementary Slackness in Pairs of Dual Systems.- 8. Extreme Points and Directions for Convex Sets.- 8.1. Extreme Points and Directions for General Convex Sets.- 8.2. Convex Hulls Revisited.- 8.3. Faces of Polyhedral Convex Sets: Extreme Points, Facets, and Edges.- 8.4. Extreme Point Representation for Polyhedral Convex Sets.- 8.5. Directions for Polyhedral Convex Sets.- 8.6. Combined Extreme Point and Extreme Direction Representation for Polyhedral Convex Sets.-8.7. Resolution of Convex Polyhedra.- 8.8. Separation of Convex Polyhedra.- 9. Simplicial Topology and Fixed Point Theorems.- 9.1. Simplexes.- 9.2. Simplicial Decomposition and Subdivision.- 9.3. Simplicial Mappings and Labeling.- 9.4. The Existence of Fixed Points.- 9.5. Fixed Points of Compact Point-to-Point Functions.- 9.6. Fixed Points of Point-to-Set Functions.- Appendix: Continuous and Hemicontinuous Functions.- References.- Notation Index.
