Plenary Lecture

A New Look at Convexity, Duality, and Optimization

Prof. Dimitri P. Bertsekas
McAfee Professor of Engineering
Massachusetts Institute of Technology
77 Massachusetts Ave.
Lab. for Information and Decision Systems
Rm 32-660D
Cambridge, MA 02139
Email: dimitrib@mit.edu
http://web.mit.edu/dimitrib/www/home.html

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.

Brief Biography of the Speaker:
Dimitri P. Bertsekas received a combined B.S.E.E. and B.S.M.E. from the National Technical University of Athens, Greece, an M.S.E.E. from George Washington University, and a Ph.D. in system science from the Massachusetts Institute of Technology in 1971.
Dr. Bertsekas has held faculty positions with the Engineering-Economic Systems Dept., Stanford University (1971-1974) and the Electrical Engineering Dept. of the University of Illinois, Urbana (1974-1979). Since 1979 he has been teaching at the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology (M.I.T.), where he is currently McAfee Professor of Engineering. He consults regularly with private industry and has held editorial positions in several journals. His research at M.I.T. spans several fields, including optimization, control, large-scale computation, and data communication networks, and is closely tied to his teaching and book authoring activities. He has written numerous research papers, and thirteen books, several of which are used as textbooks in MIT classes.
Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for his book "Neuro-Dynamic Programming" (co-authored with John Tsitsiklis), the 2000 Greek National Award for Operations Research, and the 2001 ACC John R. Ragazzini Education Award. In 2001, he was elected to the United States National Academy of Engineering.
Dr. Bertsekas' books within the last five years are "Introduction to Probability" (2002), "Convex Analysis and Optimization" (2003), "Dynamic Programming and Optimal Control: 3rd Edition" (2006), all published by Athena Scientific. He is writing a new book on Convex Optimization Theory (to appear in 2007).