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Accuracy Certificates for Convex Minimization with Inexact Oracle

Author

Listed:
  • Egor Gladin

    (HSE University
    Humboldt-Universität zu Berlin)

  • Alexander Gasnikov

    (Innopolis University
    Moscow Institute of Physics and Technology
    Institute for System Programming RAS)

  • Pavel Dvurechensky

    (Weierstrass Institute for Applied Analysis and Stochastics)

Abstract

Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy certificates produce a simple way to recover an approximate primal solution and estimate its accuracy. In this paper, we generalize accuracy certificates for the setting of an inexact first-order oracle, including the setting of primal and Lagrange dual pair of problems. We further propose an explicit way to construct accuracy certificates for a large class of cutting plane methods based on polytopes. As a by-product, we show that the considered cutting plane methods can be efficiently used with a noisy oracle even though they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in the numerical experiments highlighting that our certificates provide a tight upper bound on the objective residual.

Suggested Citation

  • Egor Gladin & Alexander Gasnikov & Pavel Dvurechensky, 2025. "Accuracy Certificates for Convex Minimization with Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-23, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02599-9
    DOI: 10.1007/s10957-024-02599-9
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    References listed on IDEAS

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    1. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, October.
    2. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," LIDAM Reprints CORE 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Kurt M. Anstreicher, 1997. "On Vaidya's Volumetric Cutting Plane Method for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 63-89, February.
    4. Arkadi Nemirovski & Shmuel Onn & Uriel G. Rothblum, 2010. "Accuracy Certificates for Computational Problems with Convex Structure," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 52-78, February.
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