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Further Study on the Convergence Rate of Alternating Direction Method of Multipliers with Logarithmic-quadratic Proximal Regularization

Author

Listed:
  • Caihua Chen

    (Nanjing University)

  • Min Li

    (Southeast University)

  • Xiaoming Yuan

    (Hong Kong Baptist University)

Abstract

In the literature, the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization has been proved to be convergent, and its worst-case convergence rate in the ergodic sense has been established. In this paper, we focus on a convex minimization model and consider an inexact version of the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization. Our primary purpose is to further study its convergence rate and to establish its worst-case convergence rates measured by the iteration complexity in both the ergodic and non-ergodic senses. In particular, existing convergence rate results for this combination are subsumed by the new results.

Suggested Citation

  • Caihua Chen & Min Li & Xiaoming Yuan, 2015. "Further Study on the Convergence Rate of Alternating Direction Method of Multipliers with Logarithmic-quadratic Proximal Regularization," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 906-929, September.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-014-0682-8
    DOI: 10.1007/s10957-014-0682-8
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Min Li & Li-Zhi Liao & Xiaoming Yuan, 2013. "Inexact Alternating Direction Methods of Multipliers with Logarithmic–Quadratic Proximal Regularization," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 412-436, November.
    3. A. Auslender & M. Teboulle, 2004. "Interior Gradient and Epsilon-Subgradient Descent Methods for Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 1-26, February.
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