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Fast Convergence of Frank-Wolfe Algorithms on Polytopes.

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Τίτλος: Fast Convergence of Frank-Wolfe Algorithms on Polytopes.
Συγγραφείς: Wirth, Elias1 (AUTHOR) wirth.elias.samuel@gmail.com, Peña, Javier2 (AUTHOR) jfp@andrew.cmu.edu, Pokutta, Sebastian1 (AUTHOR) pokutta@zib.de
Πηγή: Mathematics of Operations Research (INFORMS). May2026, Vol. 51 Issue 2, p1463-1485. 23p.
Θεματικοί όροι: *Mathematical analysis, Polytopes, Optimization algorithms, Affine transformations, Curvature
Περίληψη: We provide a template to derive affine-invariant convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound. Funding: Research reported in this paper was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—The Berlin Mathematics Research Center MATH (EXC-2046/1, project ID 390685689, BMS Stipend). Research reported in this paper was also partially supported by the Bajaj Family Chair at the Tepper School of Business, Carnegie Mellon University. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences 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: Fast Convergence of Frank-Wolfe Algorithms on Polytopes.
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Operations+Research+%28INFORMS%29%22">Mathematics of Operations Research (INFORMS)</searchLink>. May2026, Vol. 51 Issue 2, p1463-1485. 23p.
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  Data: *<searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Polytopes%22">Polytopes</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Affine+transformations%22">Affine transformations</searchLink><br /><searchLink fieldCode="DE" term="%22Curvature%22">Curvature</searchLink>
– Name: Abstract
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  Data: We provide a template to derive affine-invariant convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound. Funding: Research reported in this paper was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—The Berlin Mathematics Research Center MATH (EXC-2046/1, project ID 390685689, BMS Stipend). Research reported in this paper was also partially supported by the Bajaj Family Chair at the Tepper School of Business, Carnegie Mellon University. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences 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:
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      – Type: doi
        Value: 10.1287/moor.2024.0580
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 23
        StartPage: 1463
    Subjects:
      – SubjectFull: Mathematical analysis
        Type: general
      – SubjectFull: Polytopes
        Type: general
      – SubjectFull: Optimization algorithms
        Type: general
      – SubjectFull: Affine transformations
        Type: general
      – SubjectFull: Curvature
        Type: general
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      – TitleFull: Fast Convergence of Frank-Wolfe Algorithms on Polytopes.
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              M: 05
              Text: May2026
              Type: published
              Y: 2026
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