This talk will review a recent book treatment of convex analysis and optimization. While the subject of the book is classical, the treatment of several of its important topics is new and in some cases relies on new research. The new lines of analysis include:
(a) A unified framework for minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. Within this framework, the fundamental constraint qualifications needed for strong duality and existence of saddle points are quite apparent, and admit straightforward proofs.
(b) A unification of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. This unification is based on conditions guaranteeing that a nested family of closed convex sets has a nonempty intersection.
(c) A unification of the major constraint qualifications that guarantee the existence of Lagrange multipliers for nonconvex constrained optimization. This unification is achieved through the notion of constraint pseudonormality, which is motivated by an enhanced form of the Fritz John necessary optimality conditions.
(d) The development of incremental subgradient methods for dual optimization, and the analysis of their advantages over classical subgradient methods.
See http://www.athenasc.com/New_Look.pdf