Author | Florenzano, Monique. author |
---|---|
Title | Finite Dimensional Convexity and Optimization [electronic resource] / by Monique Florenzano, Cuong Le Van |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-3-642-56522-9 |
Descript | XII, 154 p. online resource |
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. Weylโs 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