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

M. Marques Alves, Dirk A. Lorenz, Emanuele Naldi

TL;DR

This work introduces a general framework that extends the hybrid proximal extragradient (HPE) method to degenerate preconditioned proximal point algorithms, enabling inexact, relative-error resolvent computations within splitting methods. By embedding split inclusions into a degenerate preconditioned HPE setting, the authors derive inexact versions of Douglas–Rachford, Chambolle–Pock, and Davis–Yin methods and prove weak convergence of the iterates under standard conditions. The framework unifies and recovers existing schemes (e.g., Eckstein–Yao for DR) and yields practical benefits, as demonstrated by numerical experiments where relative-error resolutions accelerate computation without sacrificing accuracy. This approach offers a flexible, scalable path for adaptive splitting in large-scale monotone inclusion problems with applications in imaging and variational problems.

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.

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

TL;DR

This work introduces a general framework that extends the hybrid proximal extragradient (HPE) method to degenerate preconditioned proximal point algorithms, enabling inexact, relative-error resolvent computations within splitting methods. By embedding split inclusions into a degenerate preconditioned HPE setting, the authors derive inexact versions of Douglas–Rachford, Chambolle–Pock, and Davis–Yin methods and prove weak convergence of the iterates under standard conditions. The framework unifies and recovers existing schemes (e.g., Eckstein–Yao for DR) and yields practical benefits, as demonstrated by numerical experiments where relative-error resolutions accelerate computation without sacrificing accuracy. This approach offers a flexible, scalable path for adaptive splitting in large-scale monotone inclusion problems with applications in imaging and variational problems.

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.
Paper Structure (24 sections, 8 theorems, 92 equations, 6 figures, 4 algorithms)

This paper contains 24 sections, 8 theorems, 92 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The hybrid proximal extragradient (HPE) method.
  3. The degenerate preconditioned proximal point method.
  4. The reduced method.
  5. Related work
  6. Contribution
  7. Notation
  8. The hybrid proximal extragradient method with degenerate preconditioning
  9. Applications to splitting methods
  10. Inexact Douglas-Rachford: retrieving Eckstein-Yao algorithm using the degenerate HPE framework
  11. Chambolle-Pock aka PDHG
  12. Inexact Davis-Yin method
  13. Experiments
  14. Chambolle-Pock
  15. Chambolle-Pock with implicit steps
  16. ...and 9 more sections

Key Result

Proposition 2.3

Let $\mathcal{A}$ be maximal monotone on ${\mathcal{H}}$, $u^{*}$ a zero of $\mathcal{A}$, $\mathcal{M}$ linear, bounded and positive semidefinite on ${\mathcal{H}}$ and $u^0\in {\mathcal{H}}$. Consider sequences $\left\{u^{k}\right\},\left\{{\tilde{u}}^{k}\right\},\left\{v^{k}\right\}$ fulfilling e Consequently we also have for $k\to\infty$

Figures (6)

  • Figure 1: Objective value over iterations and objective value over time for the first run of experiment 1 described in Section \ref{['sec:cp-experiment']}.
  • Figure 2: Objective value over iterations and objective value over time for the second run of experiment 1 described in Section \ref{['sec:cp-experiment']}.
  • Figure 3: Objective value over iterations and objective value over time for experiment 2 described in Section \ref{['sec:cp-experiment']}.
  • Figure 4: Objective value over iterations and objective value over time for the first run described in Section \ref{['sec:dy-experiment']}.
  • Figure 5: Objective value over iterations and objective value over time for the second run described in Section \ref{['sec:dy-experiment']}.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 2.1: BCLN22
  • Remark 2.2
  • Proposition 2.3: Fundamental estimates
  • proof
  • Proposition 2.4: Boundedness and weak subsequential convergence
  • proof
  • Lemma 2.5: Opial property for seminorms
  • proof
  • Theorem 2.6: Weak convergence of the iterates
  • proof
  • ...and 14 more