  An algorithm framework is proposed for minimizing nonsmooth functions. The
framework is variable-metric in that, in each iteration, a step is computed
using a symmetric positive definite matrix whose value is updated as in a
quasi-Newton scheme. However, unlike previously proposed variable-metric
algorithms for minimizing nonsmooth functions, the framework exploits
self-correcting properties made possible through BFGS-type updating. In so
doing, the framework does not overly restrict the manner in which the step
computation matrices are updated, yet the scheme is controlled well enough that
global convergence guarantees can be established. The results of numerical
experiments for a few algorithms are presented to demonstrate the
self-correcting behaviors that are guaranteed by the framework.
