A nested primal–dual iterated Tikhonov method for regularized convex optimization.
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| Title: | A nested primal–dual iterated Tikhonov method for regularized convex optimization. |
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| Authors: | Aleotti, Stefano1 (AUTHOR) saleotti@studenti.uninsubria.it, Bonettini, Silvia2 (AUTHOR) silvia.bonettini@unimore.it, Donatelli, Marco1 (AUTHOR) marco.donatelli@uninsubria.it, Prato, Marco2 (AUTHOR) marco.prato@unimore.it, Rebegoldi, Simone2 (AUTHOR) simone.rebegoldi@unimore.it |
| Source: | Computational Optimization & Applications. Jun2025, Vol. 91 Issue 2, p357-395. 39p. |
| Subjects: | Computational mathematics, Extrapolation, Algorithms |
| Abstract: | Proximal–gradient methods are widely employed tools in imaging that can be accelerated by adopting variable metrics and/or extrapolation steps. One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal–dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal–dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal–gradient step with extrapolation by performing a prefixed number of primal–dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. Furthermore, we investigate the numerical performance of our proposed method by equipping it with a scaling matrix inspired by the Iterated Tikhonov method. The numerical results show that the combination of such scaling matrices and Nesterov-like extrapolation parameters yields an effective acceleration towards the solution of the problem. [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.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 185240087 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A nested primal–dual iterated Tikhonov method for regularized convex optimization. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Aleotti%2C+Stefano%22">Aleotti, Stefano</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> saleotti@studenti.uninsubria.it</i><br /><searchLink fieldCode="AR" term="%22Bonettini%2C+Silvia%22">Bonettini, Silvia</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> silvia.bonettini@unimore.it</i><br /><searchLink fieldCode="AR" term="%22Donatelli%2C+Marco%22">Donatelli, Marco</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> marco.donatelli@uninsubria.it</i><br /><searchLink fieldCode="AR" term="%22Prato%2C+Marco%22">Prato, Marco</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> marco.prato@unimore.it</i><br /><searchLink fieldCode="AR" term="%22Rebegoldi%2C+Simone%22">Rebegoldi, Simone</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> simone.rebegoldi@unimore.it</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computational+Optimization+%26+Applications%22">Computational Optimization & Applications</searchLink>. Jun2025, Vol. 91 Issue 2, p357-395. 39p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Computational+mathematics%22">Computational mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Extrapolation%22">Extrapolation</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Proximal–gradient methods are widely employed tools in imaging that can be accelerated by adopting variable metrics and/or extrapolation steps. One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal–dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal–dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal–gradient step with extrapolation by performing a prefixed number of primal–dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. Furthermore, we investigate the numerical performance of our proposed method by equipping it with a scaling matrix inspired by the Iterated Tikhonov method. The numerical results show that the combination of such scaling matrices and Nesterov-like extrapolation parameters yields an effective acceleration towards the solution of the problem. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab 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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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10589-024-00613-4 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 39 StartPage: 357 Subjects: – SubjectFull: Computational mathematics Type: general – SubjectFull: Extrapolation Type: general – SubjectFull: Algorithms Type: general Titles: – TitleFull: A nested primal–dual iterated Tikhonov method for regularized convex optimization. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Aleotti, Stefano – PersonEntity: Name: NameFull: Bonettini, Silvia – PersonEntity: Name: NameFull: Donatelli, Marco – PersonEntity: Name: NameFull: Prato, Marco – PersonEntity: Name: NameFull: Rebegoldi, Simone IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 09266003 Numbering: – Type: volume Value: 91 – Type: issue Value: 2 Titles: – TitleFull: Computational Optimization & Applications Type: main |
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