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Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases

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  • Pauline Tan

    (Université Paris Saclay)

Abstract

In the present paper, we propose a novel convergence analysis of the alternating direction method of multipliers, based on its equivalence with the overrelaxed primal–dual hybrid gradient algorithm. We consider the smooth case, where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. Under these hypotheses, a convergence proof with an optimal parameter choice is given for the primal–dual method, which leads to convergence results for the alternating direction method of multipliers. An accelerated variant of the latter, based on a parameter relaxation, is also proposed, which is shown to converge linearly with same asymptotic rate as the primal–dual algorithm.

Suggested Citation

  • Pauline Tan, 2018. "Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 377-398, February.
    • Handle: RePEc:spr:joptap:v:176:y:2018:i:2:d:10.1007_s10957-017-1211-3
    DOI: 10.1007/s10957-017-1211-3
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    References listed on IDEAS

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    1. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
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