On the Convergence of Primal-Dual Hybrid Gradient Algorithm

The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHG's convergence was established only under some restrictive conditions on its step sizes. In this paper, we revisit PDHG's convergence in the cont...

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Bibliographic Details
Published in:SIAM journal on imaging sciences Vol. 7; no. 4; pp. 2526 - 2537
Main Authors: He, Bingsheng, You, Yanfei, Yuan, Xiaoming
Format: Journal Article
Language:English
Published: 01.01.2014
Subjects:
Algorithms
Complexity
Constants
Convergence
Image processing
Imaging
Iterative methods
Mathematical models
ISSN:1936-4954, 1936-4954
Online Access:Get full text
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Summary:The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHG's convergence was established only under some restrictive conditions on its step sizes. In this paper, we revisit PDHG's convergence in the context of a saddle-point problem and try to better understand how to choose its step sizes. More specifically, we show by an extremely simple example that PDHG is not necessarily convergent even when the step sizes are fixed as tiny constants. We then show that PDHG with constant step sizes is indeed convergent if one of the functions of the saddle-point problem is strongly convex, a condition that does hold for some variational models in imaging. With this additional condition, we also establish a worst-case convergence rate measured by the iteration complexity for PDHG with constant step sizes.
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ISSN:1936-4954
1936-4954
DOI:10.1137/140963467