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 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.

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 Hahn-Banach 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 Kuhn-Tucker Conditions.- 7.3 Value function.- Exercises.- 8. Non Convex Optimization.- 8.1 Quasi-convex functions.- 8.2 Minimization of quasi-convex 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.