Rescaling nonsmooth optimization using BFGS and Shor updates

The BFGS quasi-Newton methodology, popular for smooth minimization, has also proved surprisingly effective in nonsmooth optimization. Through a variety of simple examples and computational experiments, we explore how the BFGS matrix update improves the local metric associated with a convex function even in the absence of smoothness and without using a line search. We compare the behavior of the BFGS and Shor r-algorithm updates.