February 17, 2012, Webb 1100

We discuss the Hahn-Banach-Lagrange theorem, a generalized form of the Hahn{Banach theorem. As applications, we derive various results on the existence of linear functionals in functional analysis, on the existence of Lagrange multipliers for convex optimization problems, with an explicit sharp lower bound on the norm of the solutions (multipliers), on finite families of convex functions (leading rapidly to a minimax theorem), on the existence of subgradients of convex functions, and on the Fenchel conjugate of a convex function. We give a complete proof of Rockafellar's version of the Fenchel duality theorem, and an explicit sharp lower bound for the norm of the solutions of the Fenchel duality theorem in terms of elementary geometric concepts. The papers relevant to this talk can be found at http://www.math.ucsb.edu/~simons/WCOM.html

Stephen Simons was born in London, England. He received both his B.A. and Ph.D from Trinity College in Cambridge, England, in 1959, and 1962, respectively. He began teaching at the University of California, Santa Barbara in 1965 in the Mathematics department. Professor Simons served as the Chair of the Mathematics dept. at UCSB from 1975-77, and from 1988-89. He also served as the Assistant Dean of the College of Letters and Sciences in 1987. Professor Simons has served on the board of trustees of the Mathematical Sciences Research Institute at Berkeley, the editorial board of “Journal of Convex Analysis”, and the editorial board of “Set-Valued and Variational Analysis”. He is currently a Professor Emeritus of Math at UCSB.