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Proximal subgradient method for non-Lipschitz objective functions.

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Title: Proximal subgradient method for non-Lipschitz objective functions.
Authors: Toyoda, Mitsuru1 (AUTHOR) toyoda@tmu.ac.jp, Tanaka, Mirai2,3 (AUTHOR)
Source: Optimization Methods & Software. Aug2025, Vol. 40 Issue 4, p755-782. 28p.
Subjects: Subgradient methods, Mathematical optimization, Lipschitz continuity, Empirical research, Asymptotic analysis, Convex functions, Optimization algorithms
Abstract: This paper presents a convergence analysis framework of the proximal subgradient method for optimization problems involving non-Lipschitz continuous objective functions. In the conventional analysis of the various subgradient methods, including the proximal subgradient method, the Lipschitz continuity assumption has been placed to guarantee the boundedness of the subgradient and derive the convergence rate. However, the Lipschitz continuity does not hold in practical problems, including the sum-of- $ \ell _2 $ ℓ 2 -norms (SO $ \ell _2 $ ℓ 2 N) optimal control problem, which is examined in the numerical experiments of this paper. Without the Lipschitz continuity assumption, this paper provides the convergence analysis for strongly convex and non-strongly convex objective functions under mild assumptions. Suitable stepsize rules and resulting convergence rates are established; non-strongly convex cases result in a rate close to the rate of the existing subgradient method, and strongly convex cases achieve the same rate as the existing convergence analysis. [ABSTRACT FROM AUTHOR]
Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd 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: Proximal subgradient method for non-Lipschitz objective functions.
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  Data: <searchLink fieldCode="AR" term="%22Toyoda%2C+Mitsuru%22">Toyoda, Mitsuru</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> toyoda@tmu.ac.jp</i><br /><searchLink fieldCode="AR" term="%22Tanaka%2C+Mirai%22">Tanaka, Mirai</searchLink><relatesTo>2,3</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Optimization+Methods+%26+Software%22">Optimization Methods & Software</searchLink>. Aug2025, Vol. 40 Issue 4, p755-782. 28p.
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  Data: <searchLink fieldCode="DE" term="%22Subgradient+methods%22">Subgradient methods</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Lipschitz+continuity%22">Lipschitz continuity</searchLink><br /><searchLink fieldCode="DE" term="%22Empirical+research%22">Empirical research</searchLink><br /><searchLink fieldCode="DE" term="%22Asymptotic+analysis%22">Asymptotic analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+functions%22">Convex functions</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: This paper presents a convergence analysis framework of the proximal subgradient method for optimization problems involving non-Lipschitz continuous objective functions. In the conventional analysis of the various subgradient methods, including the proximal subgradient method, the Lipschitz continuity assumption has been placed to guarantee the boundedness of the subgradient and derive the convergence rate. However, the Lipschitz continuity does not hold in practical problems, including the sum-of- $ \ell _2 $ ℓ 2 -norms (SO $ \ell _2 $ ℓ 2 N) optimal control problem, which is examined in the numerical experiments of this paper. Without the Lipschitz continuity assumption, this paper provides the convergence analysis for strongly convex and non-strongly convex objective functions under mild assumptions. Suitable stepsize rules and resulting convergence rates are established; non-strongly convex cases result in a rate close to the rate of the existing subgradient method, and strongly convex cases achieve the same rate as the existing convergence analysis. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Optimization Methods & Software is the property of Taylor & Francis Ltd 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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    Identifiers:
      – Type: doi
        Value: 10.1080/10556788.2025.2475405
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 28
        StartPage: 755
    Subjects:
      – SubjectFull: Subgradient methods
        Type: general
      – SubjectFull: Mathematical optimization
        Type: general
      – SubjectFull: Lipschitz continuity
        Type: general
      – SubjectFull: Empirical research
        Type: general
      – SubjectFull: Asymptotic analysis
        Type: general
      – SubjectFull: Convex functions
        Type: general
      – SubjectFull: Optimization algorithms
        Type: general
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      – TitleFull: Proximal subgradient method for non-Lipschitz objective functions.
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            – D: 01
              M: 08
              Text: Aug2025
              Type: published
              Y: 2025
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