Finite Dimensional Convexity and Optimization

Overview

"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 mathematical programming whose research fields are applied mathematics and economics. The text consists of a systematic development in eight chapters, with guided exercises containing sometimes significant and useful additional results. The book is
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Paperback (Softcover reprint of the original 1st ed. 2001)
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Overview

"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 mathematical programming whose research fields are applied mathematics and economics. The text consists of a systematic development in eight chapters, with guided exercises containing sometimes significant and useful additional results. The book is appropriate as a class text, or for self-study."--BOOK JACKET.
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Product Details

  • ISBN-13: 9783642625701
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 1/28/2013
  • Series: Studies in Economic Theory Series , #13
  • Edition description: Softcover reprint of the original 1st ed. 2001
  • Edition number: 1
  • Pages: 154
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.37 (d)

Table of Contents

Notation
1 Convexity in R[superscript n] 1
1.1 Basic concepts 1
1.2 Topological properties of convex sets 8
2 Separation and Polarity 21
2.1 Separation of convex sets 21
2.2 Polars of convex sets and orthogonal subspaces 28
2.3 Application to Minkowski-Farkas' lemma 29
3 Extremal Structure of Convex Sets 37
3.1 Extreme points and faces of convex sets 37
3.2 Application to linear inequalities. Weyl's theorem 40
3.3 Extreme points and extremal subsets of a polyhedral convex set 45
4 Linear Programming 51
4.1 Necessary and sufficient conditions of optimality 52
4.2 The duality theorem of linear programming 54
4.3 The simplex method 60
5 Convex Functions 73
5.1 Basic definitions and properties 73
5.2 Continuity theorems 76
5.3 Continuity properties of collections of convex functions 79
6 Differential Theory of Convex Functions 87
6.1 The Hahn-Banach dominated extension theorem 88
6.2 Sublinear functions 89
6.3 Support functions 91
6.4 Directional derivatives 92
6.5 Subgradients and subdifferential of a convex function 93
6.6 Differentiability of convex functions 98
6.7 Differential continuity for convex functions 100
7 Convex Optimization With Convex Constraints 109
7.1 The minimum of a convex function f: R[superscript n] [approaches]R 110
7.2 Kuhn-Tucker Conditions 111
7.3 Value functions 123
8 Non Convex Optimization 129
8.1 Quasi-convex functions 129
8.2 Minimization of quasi-convex functions 134
8.3 Differentiable optimization 137
A Appendix 141
Bibliography 147
Index 151
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