Inertial Quasi-Newton Methods for Monotone Inclusion: Efficient Resolvent Calculus and Primal-Dual Methods

TL;DR

This paper develops an inertial quasi-Newton algorithm for monotone inclusions, introducing an efficient resolvent calculus for low-rank metrics and demonstrating improved convergence and performance in image processing tasks.

Contribution

It introduces a novel inertial quasi-Newton method with an efficient resolvent calculus for low-rank metrics, enhancing convergence for monotone inclusions and saddle point problems.

Findings

Proves convergence and linear rates under strong monotonicity.

Develops a new resolvent calculus for quasi-Newton metrics.

Demonstrates improved performance in image processing experiments.

Abstract

We introduce an inertial quasi-Newton Forward-Backward Splitting Algorithm to solve a class of monotone inclusion problems. While the inertial step is computationally cheap, in general, the bottleneck is the evaluation of the resolvent operator. A change of the metric makes its computation hard even for (otherwise in the standard metric) simple operators. In order to fully exploit the advantage of adapting the metric, we develop a new efficient resolvent calculus for a low-rank perturbed standard metric, which accounts exactly for quasi-Newton metrics. Moreover, we prove the convergence of our algorithms, including linear convergence rates in case one of the two considered operators is strongly monotone. Beyond the general monotone inclusion setup, we instantiate a novel inertial quasi-Newton Primal-Dual Hybrid Gradient Method for solving saddle point problems. The favourable performance of our inertial quasi-Newton PDHG method is demonstrated on several numerical experiments in image processing.