On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications
In this paper, we study the nonexpansive properties of metric resolvent and present the convergence analysis for the associated fixed-point iterations of both Banach–Picard and Krasnosel’skiĭ–Mann types. A by-product of our expositions also extends the proximity operator and Moreau’s decomposition i...
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| Veröffentlicht in: | Journal of the Operations Research Society of China (Internet) Jg. 13; H. 4; S. 966 - 988 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.12.2025
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| Abstract | In this paper, we study the nonexpansive properties of metric resolvent and present the convergence analysis for the associated fixed-point iterations of both Banach–Picard and Krasnosel’skiĭ–Mann types. A by-product of our expositions also extends the proximity operator and Moreau’s decomposition identity to arbitrary metric. It is further shown that many classes of the first-order operator splitting algorithms, including the alternating direction methods of multipliers, primal–dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be easily obtained within this unified framework. |
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| AbstractList | In this paper, we study the nonexpansive properties of metric resolvent and present the convergence analysis for the associated fixed-point iterations of both Banach–Picard and Krasnosel’skiĭ–Mann types. A by-product of our expositions also extends the proximity operator and Moreau’s decomposition identity to arbitrary metric. It is further shown that many classes of the first-order operator splitting algorithms, including the alternating direction methods of multipliers, primal–dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be easily obtained within this unified framework. |
| Author | Xue, Feng |
| Author_xml | – sequence: 1 givenname: Feng surname: Xue fullname: Xue, Feng email: fxue@link.cuhk.edu.hk organization: National Key Laboratory |
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| Cites_doi | 10.1016/j.na.2012.09.008 10.1137/040605412 10.1007/s10851-017-0709-5 10.1007/s10107-018-1284-2 10.1090/S0025-5718-09-02242-X 10.1007/s10107-014-0826-5 10.1007/s00211-014-0673-6 10.1007/s40314-016-0371-3 10.1051/m2an/197509R200411 10.1007/978-3-662-12613-4 10.1007/s10107-014-0805-x 10.1137/09076934X 10.1007/s10107-015-0964-4 10.1561/2200000016 10.1007/s12532-015-0078-2 10.1007/s10851-010-0251-1 10.1137/100814494 10.1007/s10107-014-0766-0 10.1007/978-3-319-48311-5 10.1137/110836936 10.1137/070688146 10.1109/CDC.2014.7040175 10.1007/s10957-017-1112-5 10.1007/s10898-016-0405-9 10.1137/050626090 10.1137/1.9781611974997 10.1137/080725891 10.1007/s11228-017-0421-z 10.1007/s00025-022-01766-6 10.1090/S0002-9904-1967-11761-0 10.1007/s10107-014-0850-5 10.1137/13090849X 10.1137/S0363012992235547 10.1137/070703983 10.1137/120872802 10.1137/100818327 10.1109/TAC.2016.2564160 10.1090/S0025-5718-08-02189-3 10.1007/s10589-018-9994-1 10.1137/0314056 10.1007/s10915-016-0318-2 10.1109/TMI.2014.2321098 10.1007/s10915-010-9408-8 10.1007/s10915-017-0477-9 10.1090/S0025-5718-2012-02598-1 10.1137/0716071 10.1016/0022-247X(79)90234-8 |
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| SubjectTerms | Algorithms Convergence Convex analysis Decomposition Management Science Mathematics Mathematics and Statistics Operations Research |
| Subtitle | On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications |
| Title | On the Metric Resolvent: Nonexpansiveness, Convergence Rates and Applications |
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