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

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Title: A general framework for inexact splitting algorithms with relative errors and applications to Chambolle–Pock and Davis–Yin methods.
Authors: Alves, M. Marques1 (AUTHOR) maicon.alves@ufsc.br, Lorenz, Dirk A.2 (AUTHOR) d.lorenz@uni-bremen.de, Naldi, Emanuele3 (AUTHOR) emanuele.naldi@edu.unige.it
Source: Computational Optimization & Applications. Mar2026, Vol. 93 Issue 2, p729-763. 35p.
Subjects: Monotone operators, Optimization algorithms, Approximation error
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. [ABSTRACT FROM AUTHOR]
Copyright of Computational Optimization & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Optimization & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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      – SubjectFull: Approximation error
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              Text: Mar2026
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