Finite Dimensional Convexity and Optimization
The primary aim of this book is to present notions of convex analysis which constitute the basic underlying structure of argumentation in economic theory and which are common to optimization problems encountered in many applications. The intended readers are graduate students, and specialists of mat...
Main Authors:  , 

Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
2001, 2001

Edition:  1st ed. 2001 
Series:  Studies in Economic Theory

Subjects:  
Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 1. Convexity in ?n
 1.1 Basic concepts
 1.2. Topological properties of convex sets
 Exercises
 2. Separation and Polarity
 2.1 Separation of convex sets
 2.2 Polars of convex sets and orthogonal subspaces
 Exercises
 3. Extremal Structure of Convex Sets
 3.1 Extreme points and faces of convex sets
 3.2 Application to linear inequalities. Weyl’s theorem
 3.3 Extreme points and extremal subsets of a polyhedral convex set
 Exercises
 4. Linear Programming
 4.1 Necessary and sufficient conditions of optimality
 4.2 The duality theorem of linear programming
 4.3 The simplex method
 Exercises
 5. Convex Functions
 5.1 Basic definitions and properties
 5.2 Continuity theorems
 5.3 Continuity properties of collections of convex functions
 Exercises
 6. Differential Theory of Convex Functions
 6.1 The HahnBanach dominated extension theorem
 6.2 Sublinear functions
 6.3 Support functions
 6.4 Directional derivatives
 6.5 Subgradients and subdifferential of a convex function
 6.6 Differentiability of convex functions
 6.7 Differential continuity for convex functions
 Exercises
 7. Convex Optimization With Convex Constraints
 7.1 The minimum of a convex function f: ?n ? ?
 7.2 KuhnTucker Conditions
 7.3 Value function
 Exercises
 8. Non Convex Optimization
 8.1 Quasiconvex functions
 8.2 Minimization of quasiconvex functions
 8.3 Differentiate optimization
 Exercises
 A. Appendix
 A.1 Some preliminaries on topology
 A.2 The Mean value theorem
 A.3 The Local inversion theorem
 A.4 The implicit functions theorem