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

The generalized proximal point algorithm with step size 2 is not necessarily convergent 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Optimization and Applications Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Optimization; Operations Research, Management Science; Operations Research/Decision Theory; Statistics, general; Convex and Discrete Geometry
ISSN
0926-6003
eISSN
1573-2894
D.O.I.
10.1007/s10589-018-9992-3
Publisher site
See Article on Publisher Site

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.

Journal

Computational Optimization and ApplicationsSpringer Journals

Published: Mar 3, 2018

References

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