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A generic algorithmic framework is proposed for minimizing nonsmooth and potentially nonconvex objective functions. The framework is variable-metric in the sense that, in each iteration, a step is computed using a symmetric positive definite matrix whose value is updated in a similar manner as in the Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme popular for minimizing smooth objectives. Unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the proposed framework exploits the self-correcting properties of 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 various algorithms are presented to demonstrate the self-correcting behaviors that are guaranteed by the framework.

A self-correcting variable-metric algorithm framework for nonsmooth optimization