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Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

Author

Listed:
  • Max L. N. Gonçalves

    (Universidade Federal de Goiás)

  • Maicon Marques Alves

    (Universidade Federal de Santa Catarina)

  • Jefferson G. Melo

    (Universidade Federal de Goiás)

Abstract

In this paper, we obtain global pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multipliers for solving linearly constrained convex optimization problems. We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient framework for solving monotone inclusions. Then, the convergence rates for the former method are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the variable metric proximal alternating direction method of multipliers (resp. variable metric hybrid proximal extragradient framework).

Suggested Citation

  • Max L. N. Gonçalves & Maicon Marques Alves & Jefferson G. Melo, 2018. "Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 448-478, May.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:2:d:10.1007_s10957-018-1232-6
    DOI: 10.1007/s10957-018-1232-6
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    References listed on IDEAS

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    1. B. S. He & H. Yang & S. L. Wang, 2000. "Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 337-356, August.
    2. Ying Cui & Xudong Li & Defeng Sun & Kim-Chuan Toh, 2016. "On the Convergence Properties of a Majorized Alternating Direction Method of Multipliers for Linearly Constrained Convex Optimization Problems with Coupled Objective Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1013-1041, June.
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    Cited by:

    1. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2019. "Iteration-complexity analysis of a generalized alternating direction method of multipliers," Journal of Global Optimization, Springer, vol. 73(2), pages 331-348, February.
    2. Vando A. Adona & Max L. N. Gonçalves & Jefferson G. Melo, 2019. "A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 640-666, August.
    3. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2020. "An inexact proximal generalized alternating direction method of multipliers," Computational Optimization and Applications, Springer, vol. 76(3), pages 621-647, July.

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