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Article: On the convergence of primal-dual hybrid gradient algorithm
Title | On the convergence of primal-dual hybrid gradient algorithm |
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Authors | |
Keywords | Primal-dual hybrid gradient algorithm Saddle-point problem Total variation Convergence rate Image restoration Convex optimization |
Issue Date | 2014 |
Publisher | Society for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siims.php |
Citation | SIAM Journal on Imaging Sciences, 2014, v. 7, n. 4, p. 2526-2537 How to Cite? |
Abstract | © 2014 Society for Industrial and Applied Mathematics. The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHGâ s convergence was established only under some restrictive conditions on its step sizes. In this paper, we revisit PDHGâ 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. |
Persistent Identifier | http://hdl.handle.net/10722/251082 |
ISSN | 2023 Impact Factor: 2.1 2023 SCImago Journal Rankings: 0.960 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | He, Bingsheng | - |
dc.contributor.author | You, Yanfei | - |
dc.contributor.author | Yuan, Xiaoming | - |
dc.date.accessioned | 2018-02-01T01:54:31Z | - |
dc.date.available | 2018-02-01T01:54:31Z | - |
dc.date.issued | 2014 | - |
dc.identifier.citation | SIAM Journal on Imaging Sciences, 2014, v. 7, n. 4, p. 2526-2537 | - |
dc.identifier.issn | 1936-4954 | - |
dc.identifier.uri | http://hdl.handle.net/10722/251082 | - |
dc.description.abstract | © 2014 Society for Industrial and Applied Mathematics. The primal-dual hybrid gradient algorithm (PDHG) has been widely used, especially for some basic image processing models. In the literature, PDHGâ s convergence was established only under some restrictive conditions on its step sizes. In this paper, we revisit PDHGâ 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. | - |
dc.language | eng | - |
dc.publisher | Society for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/siims.php | - |
dc.relation.ispartof | SIAM Journal on Imaging Sciences | - |
dc.subject | Primal-dual hybrid gradient algorithm | - |
dc.subject | Saddle-point problem | - |
dc.subject | Total variation | - |
dc.subject | Convergence rate | - |
dc.subject | Image restoration | - |
dc.subject | Convex optimization | - |
dc.title | On the convergence of primal-dual hybrid gradient algorithm | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1137/140963467 | - |
dc.identifier.scopus | eid_2-s2.0-84919665374 | - |
dc.identifier.volume | 7 | - |
dc.identifier.issue | 4 | - |
dc.identifier.spage | 2526 | - |
dc.identifier.epage | 2537 | - |
dc.identifier.isi | WOS:000346854900021 | - |
dc.identifier.issnl | 1936-4954 | - |