A Self-Correcting Variable-Metric Algorithm Framework for Nonsmooth Optimization

TL;DR

This paper introduces a novel variable-metric algorithm framework for nonsmooth optimization that leverages self-correcting BFGS-type updates to ensure convergence and improve robustness.

Contribution

It presents a new self-correcting variable-metric framework for nonsmooth optimization that guarantees global convergence using flexible BFGS-type updates.

Findings

Framework demonstrates self-correcting properties in numerical experiments.

Convergence guarantees are established for the proposed scheme.

Numerical results show improved robustness over existing methods.

Abstract

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.