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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| Vydáno v: | SIAM journal on imaging sciences Ročník 7; číslo 4; s. 2526 - 2537 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.01.2014
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| ISSN: | 1936-4954, 1936-4954 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Yuan, Xiaoming You, Yanfei He, Bingsheng |
| Author_xml | – sequence: 1 givenname: Bingsheng surname: He fullname: He, Bingsheng – sequence: 2 givenname: Yanfei surname: You fullname: You, Yanfei – sequence: 3 givenname: Xiaoming surname: Yuan fullname: Yuan, Xiaoming |
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| Cites_doi | 10.1137/080725891 10.1137/080733371 10.1137/09076934X 10.1137/070696143 10.1016/0167-2789(92)90242-F 10.1093/imanum/22.3.329 10.1023/B:JMIV.0000011320.81911.38 10.1007/s10851-011-0324-9 10.1137/100814494 10.1007/BF00927673 10.1007/s10851-010-0251-1 10.1007/s10107-012-0629-5 10.1007/s10915-010-9408-8 10.1090/S0025-5718-08-02189-3 10.3934/ipi.2012.6.357 10.1137/080738970 |
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