Finite Dimensional Convexity and Optimization

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.