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Convergence rates for an inexact ADMM applied to separable convex optimization

Author

Listed:
  • William W. Hager

    (University of Florida)

  • Hongchao Zhang

    (Louisiana State University)

Abstract

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k, the convergence rate is $$\mathcal{{O}}(1/k)$$ O ( 1 / k ) in a convex setting and $$\mathcal{{O}}(1/k^2)$$ O ( 1 / k 2 ) in a strongly convex setting. When an error bound condition holds, the algorithm is 2-step linearly convergent. The I-ADMM is designed so that the accuracy of the inexact iteration preserves the global convergence rates of the exact iteration, leading to better numerical performance in the test problems.

Suggested Citation

  • William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    • Handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00221-y
    DOI: 10.1007/s10589-020-00221-y
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    References listed on IDEAS

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    Cited by:

    1. Hélène Le Cadre & Yuting Mou & Hanspeter Höschle, 2020. "Parametrized Inexact-ADMM to Span the Set of Generalized Nash Equilibria: A Normalized Equilibrium Approach," Working Papers hal-02925005, HAL.
    2. Shengjie Xu & Bingsheng He, 2021. "A parallel splitting ALM-based algorithm for separable convex programming," Computational Optimization and Applications, Springer, vol. 80(3), pages 831-851, December.
    3. Jianchao Bai & William W. Hager & Hongchao Zhang, 2022. "An inexact accelerated stochastic ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 81(2), pages 479-518, March.

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