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Convergence rate of inexact augmented Lagrangian method with practical relative error criterion for composite convex programming

Author

Listed:
  • Yunfei Qu

    (Beihang University)

  • Xingju Cai

    (Nanjing Normal University)

  • Hongying Liu

    (Beihang University)

  • Deren Han

    (Beihang University)

Abstract

In this paper, we consider the composite convex optimization problem with a linear equality constraint. We propose a practical inexact augmented Lagrangian (IAL) framework that employs two relative error criteria. Under the first criterion, we demonstrate convergence and establish sublinear ergodic convergence rates. By incorporating the second criterion, we achieve sublinear non-ergodic convergence rates. Furthermore, we determine the total iteration complexity of the IAL framework by slightly relaxing these criteria. Numerical experiments on both synthetic and real-world problems are conducted to illustrate the efficiency of the proposed IAL method.

Suggested Citation

  • Yunfei Qu & Xingju Cai & Hongying Liu & Deren Han, 2025. "Convergence rate of inexact augmented Lagrangian method with practical relative error criterion for composite convex programming," Computational Optimization and Applications, Springer, vol. 91(3), pages 1227-1261, July.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:3:d:10.1007_s10589-025-00683-y
    DOI: 10.1007/s10589-025-00683-y
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    References listed on IDEAS

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    1. Xin-Yuan Zhao & Liang Chen, 2020. "The Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 37(04), pages 1-16, August.
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    5. Ya-Feng Liu & Xin Liu & Shiqian Ma, 2019. "On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 632-650, May.
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