A general framework for inexact splitting algorithms with relative errors and applications to Chambolle-Pock and Davis-Yin methods

TL;DR

This paper develops a unified inexact splitting algorithm framework with relative error control, extending existing methods like Chambolle-Pock and Davis-Yin, leading to improved computational efficiency and accuracy.

Contribution

It introduces a general inexact splitting framework using degenerate preconditioners, applicable to various methods including Chambolle-Pock and Davis-Yin, with adaptive error control.

Findings

Inexact resolvent computations improve time efficiency.

The framework applies to multiple splitting methods.

Enhanced accuracy with controlled inexactness.

Abstract

In this work we apply the recently introduced framework of degenerate preconditioned proximal point algorithms to the hybrid proximal extragradient (HPE) method for maximal monotone inclusions. The latter is a method that allows inexact proximal (or resolvent) steps where the error is controlled by a relative-error criterion. Recently the HPE framework has been extended to the Douglas-Rachford method by Eckstein and Yao. In this paper we further extend the applicability of the HPE framework to splitting methods. To this end we use the framework of degenerate preconditioners that allows to write a large class of splitting methods as preconditioned proximal point algorithms. In this way, we modify many splitting methods such that one or more of the resolvents can be computed inexactly with an error that is controlled by an adaptive criterion. Further, we illustrate the algorithmic framework in the case of Chambolle-Pock's primal dual hybrid gradient method and the Davis-Yin's forward Douglas-Rachford method. In both cases, the inexact computation of the resolvent shows clear advantages in computing time and accuracy.