http://wseas.org/conferences/2007/greece/
A New Look at Convexity, Duality, and Optimization
Prof. Dimitri P. Bertsekas
McAfee Professor of Engineering
MIT (Massachusetts Institute of Technology)
77 Massachusetts Ave.
Lab. for Information and Decision Systems
Rm 32-660D
Cambridge, MA 02139
Abstract: 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.
