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First-Order Algorithms for Robust Optimization Problems via Convex-Concave Saddle-Point Lagrangian Reformulation.

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Titel: First-Order Algorithms for Robust Optimization Problems via Convex-Concave Saddle-Point Lagrangian Reformulation.
Autoren: Postek, Krzysztof, Shtern, Shimrit
Quelle: INFORMS Journal on Computing; May/Jun2025, Vol. 37 Issue 3, p557-581, 25p
Schlagwörter: ROBUST optimization, MATHEMATICAL optimization, CONVEX programming, DETERMINISTIC algorithms, AFFINE geometry
Abstract: Robust optimization (RO) is one of the key paradigms for solving optimization problems affected by uncertainty. Two principal approaches for RO, the robust counterpart method and the adversarial approach, potentially lead to excessively large optimization problems. For that reason, first-order approaches, based on online convex optimization, have been proposed as alternatives for the case of large-scale problems. However, existing first-order methods are either stochastic in nature or involve a binary search for the optimal value. We show that this problem can also be solved with deterministic first-order algorithms based on a saddle-point Lagrangian reformulation that avoids both of these issues. Our approach recovers the other approaches' O(1/ϵ2) convergence rate in the general case and offers an improved O(1/ϵ) rate for problems with constraints that are affine both in the decision and in the uncertainty. Experiment involving robust quadratic optimization demonstrates the numerical benefits of our approach. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Veni.191E.035] and the Israel Science Foundation [Grant 1460/19]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0200) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0200). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]
Copyright of INFORMS Journal on Computing 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: First-Order Algorithms for Robust Optimization Problems via Convex-Concave Saddle-Point Lagrangian Reformulation.
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  Data: <searchLink fieldCode="AR" term="%22Postek%2C+Krzysztof%22">Postek, Krzysztof</searchLink><br /><searchLink fieldCode="AR" term="%22Shtern%2C+Shimrit%22">Shtern, Shimrit</searchLink>
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  Data: INFORMS Journal on Computing; May/Jun2025, Vol. 37 Issue 3, p557-581, 25p
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  Data: <searchLink fieldCode="DE" term="%22ROBUST+optimization%22">ROBUST optimization</searchLink><br /><searchLink fieldCode="DE" term="%22MATHEMATICAL+optimization%22">MATHEMATICAL optimization</searchLink><br /><searchLink fieldCode="DE" term="%22CONVEX+programming%22">CONVEX programming</searchLink><br /><searchLink fieldCode="DE" term="%22DETERMINISTIC+algorithms%22">DETERMINISTIC algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22AFFINE+geometry%22">AFFINE geometry</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Robust optimization (RO) is one of the key paradigms for solving optimization problems affected by uncertainty. Two principal approaches for RO, the robust counterpart method and the adversarial approach, potentially lead to excessively large optimization problems. For that reason, first-order approaches, based on online convex optimization, have been proposed as alternatives for the case of large-scale problems. However, existing first-order methods are either stochastic in nature or involve a binary search for the optimal value. We show that this problem can also be solved with deterministic first-order algorithms based on a saddle-point Lagrangian reformulation that avoids both of these issues. Our approach recovers the other approaches' O(1/ϵ2) convergence rate in the general case and offers an improved O(1/ϵ) rate for problems with constraints that are affine both in the decision and in the uncertainty. Experiment involving robust quadratic optimization demonstrates the numerical benefits of our approach. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant VI.Veni.191E.035] and the Israel Science Foundation [Grant 1460/19]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0200) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0200). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of INFORMS Journal on Computing 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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        Value: 10.1287/ijoc.2022.0200
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      – Code: eng
        Text: English
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        PageCount: 25
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        Type: general
      – SubjectFull: MATHEMATICAL optimization
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      – SubjectFull: CONVEX programming
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      – SubjectFull: DETERMINISTIC algorithms
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      – SubjectFull: AFFINE geometry
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      – TitleFull: First-Order Algorithms for Robust Optimization Problems via Convex-Concave Saddle-Point Lagrangian Reformulation.
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              Text: May/Jun2025
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