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

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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.

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Notes

  1. Note that it is even possible to consider \(\gamma >2\) in (4); but this case usually requires much stronger assumptions on T to ensure the convergence, see, e.g., [7] for details. We thus do not consider the case of \(\gamma >2\) in this note.

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Authors and Affiliations

  1. Department of Mathematics, National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210093, China

    Min Tao

  2. The University of Hong Kong, Hong Kong, China

    Xiaoming Yuan

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  1. Min Tao
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  2. Xiaoming Yuan
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Correspondence to Min Tao.

Additional information

Min Tao was supported by the Natural Science Foundation of China: NSFC-11301280 and the sponsorship of Jiangsu overseas research and training program for university prominent young and middle-aged teachers and presidents. Xiaoming Yuan was supported by the General Research Fund from Hong Kong Research Grants Council: HKBU 12313516.

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Tao, M., Yuan, X. The generalized proximal point algorithm with step size 2 is not necessarily convergent. Comput Optim Appl 70, 827–839 (2018). https://doi.org/10.1007/s10589-018-9992-3

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