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A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis

Author

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  • Vando A. Adona

    (Universidade Federal de Goiás)

  • Max L. N. Gonçalves

    (Universidade Federal de Goiás)

  • Jefferson G. Melo

    (Universidade Federal de Goiás)

Abstract

This paper proposes a partially inexact proximal alternating direction method of multipliers for computing approximate solutions of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact alternating direction method of multipliers with relative error criteria have been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:2:d:10.1007_s10957-019-01525-8
    DOI: 10.1007/s10957-019-01525-8
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    References listed on IDEAS

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    1. 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.
    2. Jonathan Eckstein & Wang Yao, 2017. "Approximate ADMM algorithms derived from Lagrangian splitting," Computational Optimization and Applications, Springer, vol. 68(2), pages 363-405, November.
    3. Gonçalves, M.L.N., 2018. "On the pointwise iteration-complexity of a dynamic regularized ADMM with over-relaxation stepsize," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 315-325.
    4. M. H. Xu, 2007. "Proximal Alternating Directions Method for Structured Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 107-117, July.
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    Cited by:

    1. Yunier Bello-Cruz & Max L. N. Gonçalves & Nathan Krislock, 2023. "On FISTA with a relative error rule," Computational Optimization and Applications, Springer, vol. 84(2), pages 295-318, March.
    2. 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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