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The generalized proximal point algorithm with step size 2 is not necessarily convergent

Author

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  • Min Tao

    (Nanjing University)

  • Xiaoming Yuan

    (The University of Hong Kong)

Abstract

The proximal point algorithm (PPA) is a fundamental method in optimization and it has been well studied in the literature. Recently a generalized version of the PPA with a step size in (0, 2) has been proposed. Inheriting all important theoretical properties of the original PPA, the generalized PPA has some numerical advantages that have been well verified in the literature by various applications. A common sense is that larger step sizes are preferred whenever the convergence can be theoretically ensured; thus it is interesting to know whether or not the step size of the generalized PPA can be as large as 2. We give a negative answer to this question. Some counterexamples are constructed to illustrate the divergence of the generalized PPA with step size 2 in both generic and specific settings, including the generalized versions of the very popular augmented Lagrangian method and the alternating direction method of multipliers. A by-product of our analysis is the failure of convergence of the Peaceman–Rachford splitting method and a generalized version of the forward–backward splitting method with step size 1.5.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9992-3
    DOI: 10.1007/s10589-018-9992-3
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.

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