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On Glowinski’s Open Question on the Alternating Direction Method of Multipliers

Author

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  • Min Tao

    (Nanjing University)

  • Xiaoming Yuan

    (The University of Hong Kong)

Abstract

The alternating direction method of multipliers was proposed by Glowinski and Marrocco in 1974, and it has been widely used in a broad spectrum of areas, especially in some sparsity-driven application domains. In 1982, Fortin and Glowinski suggested to enlarge the range of the dual step size for updating the multiplier from 1 to the open interval of zero to the golden ratio, and this strategy immediately accelerates the convergence of alternating direction method of multipliers for most of its applications. Meanwhile, Glowinski raised the question of whether or not the range can be further enlarged to the open interval of zero to 2; this question remains open with nearly no progress in the past decades. In this paper, we answer this question affirmatively for the case where both the functions in the objective function are quadratic. Thus, Glowinski’s open question is partially answered. We further establish the global linear convergence of the alternating direction method of multipliers with this enlarged step size range for the quadratic programming under a tight condition.

Suggested Citation

  • Min Tao & Xiaoming Yuan, 2018. "On Glowinski’s Open Question on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 163-196, October.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:1:d:10.1007_s10957-018-1338-x
    DOI: 10.1007/s10957-018-1338-x
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    References listed on IDEAS

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    1. Min Tao & Xiaoming Yuan, 2018. "The generalized proximal point algorithm with step size 2 is not necessarily convergent," Computational Optimization and Applications, Springer, vol. 70(3), pages 827-839, July.
    2. Sun, Jie & Zhang, Su, 2010. "A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1210-1220, December.
    3. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    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. Min Tao, 2020. "Convergence study of indefinite proximal ADMM with a relaxation factor," Computational Optimization and Applications, Springer, vol. 77(1), pages 91-123, September.

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