Convex Analysis

Convex sets in IR and their basic properties, separation of convex sets, properties of convex polyhedra (and polytones). Convex functions continuity and differentiability properties, subdifferentiability, duality of convex sets, Fenchel dual of a convex function, bipolar theorem. Convex programming, dual convex programs, perturbation and Lagrangian approaches to duality, the connection between the two approaches, saddle point theorems. Applications of convex analysis: inequalities, interior-point methods, approximation, merit functions.