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

Min Tao () and Xiaoming Yuan ()
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Min Tao: Nanjing University
Xiaoming Yuan: The University of Hong Kong

Computational Optimization and Applications, 2018, vol. 70, issue 3, No 7, 827-839

Abstract: 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.

Keywords: Proximal point algorithm; Step size; Convergence; Augmented Lagrangian method; Alternating direction method of multipliers; Peaceman–Rachford splitting method; Forward–backward splitting method; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10589-018-9992-3

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