Colloquium - 9/23, 3:00PM, Duncan Hall 1064

Richard A. Tapia

Department of Computational and Applied Mathematics
Rice University

"In Search of the Optimizers Holy Grail: Theory Establishing the Numerical Superiority of the BFGs Secant Update"

In this talk we present several representation theorems and averaging theorems for members of the difference class of secant updates introduced by Brodlie, Gourlay, and Greenstadt in 1973. Major contributions are that the integral form of the mean-value theorem leads to a proof that the BFGS update is pointwise the infinite average of all the updates on the one-dimension manifold in the Dennis class that connects the DFP secant update to the Greenstadt update, and that it can be expressed as the pointwise average of these latter two updates. Analogous results hold for all secant updates that belong to the difference class. While we gain new understanding of the structural properties of the highly popular BFGS secant update and other updates from the difference class; these results lead us to the belief that the Holy Grail does not exist.