Convergence of the preconditioned proximal point method and Douglas–Rachford splitting in the absence of monotonicity.
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| Title: | Convergence of the preconditioned proximal point method and Douglas–Rachford splitting in the absence of monotonicity. |
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| Authors: | Evens, Brecht1 (AUTHOR) brecht.evens@kuleuven.be, Pas, Pieter1 (AUTHOR) pieter.pas@kuleuven.be, Latafat, Puya1 (AUTHOR) puya.latafat@kuleuven.be, Patrinos, Panagiotis1 (AUTHOR) panos.patrinos@kuleuven.be |
| Source: | Mathematical Programming. Nov2025, Vol. 214 Issue 1/2, p247-301. 55p. |
| Subjects: | Numerical analysis, Monotone operators, Optimization algorithms, Mathematical analysis, Nonlinear programming |
| Abstract: | The proximal point algorithm (PPA) is the most widely recognized method for solving inclusion problems and serves as the foundation for many numerical algorithms. Despite this popularity, its convergence results have been largely limited to the monotone setting. In this work, we study the convergence of (relaxed) preconditioned PPA for a class of nonmonotone problems that satisfy an oblique weak Minty condition. Additionally, we study the (relaxed) Douglas-Rachford splitting (DRS) method in the nonmonotone setting by establishing a connection between DRS and the preconditioned PPA with a positive semidefinite preconditioner. To better characterize the class of problems covered by our analysis, we introduce the class of semimonotone operators, offering a natural extension to (hypo)monotone and co(hypo)monotone operators, and describe some of their properties. Sufficient conditions for global convergence of DRS involving the sum of two semimonotone operators are provided. Notably, it is shown that DRS converges even when the sum of the involved operators (or of their inverses) is nonmonotone. Various example problems are provided, demonstrating the tightness of our convergence results and highlighting the wide range of applications our theory is able to cover. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematical Programming 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: 189704698 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Convergence of the preconditioned proximal point method and Douglas–Rachford splitting in the absence of monotonicity. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Evens%2C+Brecht%22">Evens, Brecht</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> brecht.evens@kuleuven.be</i><br /><searchLink fieldCode="AR" term="%22Pas%2C+Pieter%22">Pas, Pieter</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> pieter.pas@kuleuven.be</i><br /><searchLink fieldCode="AR" term="%22Latafat%2C+Puya%22">Latafat, Puya</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> puya.latafat@kuleuven.be</i><br /><searchLink fieldCode="AR" term="%22Patrinos%2C+Panagiotis%22">Patrinos, Panagiotis</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> panos.patrinos@kuleuven.be</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Mathematical+Programming%22">Mathematical Programming</searchLink>. Nov2025, Vol. 214 Issue 1/2, p247-301. 55p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Monotone+operators%22">Monotone operators</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+programming%22">Nonlinear programming</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: The proximal point algorithm (PPA) is the most widely recognized method for solving inclusion problems and serves as the foundation for many numerical algorithms. Despite this popularity, its convergence results have been largely limited to the monotone setting. In this work, we study the convergence of (relaxed) preconditioned PPA for a class of nonmonotone problems that satisfy an oblique weak Minty condition. Additionally, we study the (relaxed) Douglas-Rachford splitting (DRS) method in the nonmonotone setting by establishing a connection between DRS and the preconditioned PPA with a positive semidefinite preconditioner. To better characterize the class of problems covered by our analysis, we introduce the class of semimonotone operators, offering a natural extension to (hypo)monotone and co(hypo)monotone operators, and describe some of their properties. Sufficient conditions for global convergence of DRS involving the sum of two semimonotone operators are provided. Notably, it is shown that DRS converges even when the sum of the involved operators (or of their inverses) is nonmonotone. Various example problems are provided, demonstrating the tightness of our convergence results and highlighting the wide range of applications our theory is able to cover. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Mathematical Programming 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/s10107-024-02182-0 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 55 StartPage: 247 Subjects: – SubjectFull: Numerical analysis Type: general – SubjectFull: Monotone operators Type: general – SubjectFull: Optimization algorithms Type: general – SubjectFull: Mathematical analysis Type: general – SubjectFull: Nonlinear programming Type: general Titles: – TitleFull: Convergence of the preconditioned proximal point method and Douglas–Rachford splitting in the absence of monotonicity. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Evens, Brecht – PersonEntity: Name: NameFull: Pas, Pieter – PersonEntity: Name: NameFull: Latafat, Puya – PersonEntity: Name: NameFull: Patrinos, Panagiotis IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: Nov2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 00255610 Numbering: – Type: volume Value: 214 – Type: issue Value: 1/2 Titles: – TitleFull: Mathematical Programming Type: main |
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