Practical proximal primal-dual algorithms for structured saddle point problems Practical proximal primal-dual algorithms for structured saddle point problems

In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Pro...

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Published in:	Journal of global optimization Vol. 93; no. 3; pp. 803 - 831
Main Authors:	Qu, Yunfei, He, Hongjin, Zhang, Tao, Han, Deren
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.11.2025 Springer Nature B.V
Subjects:	Algorithms Computer Science Convex analysis Euclidean space Lipschitz condition Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Saddle points restarting technique Primary 65K05 saddle point problem inexact criterion Primal-dual algorithm 90C25 65K10 accelerated algorithm
ISSN:	0925-5001, 1573-2916
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Abstract	In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Proximal Primal-Dual algorithm (PPD+), which possesses an O ( 1 / N 2 ) convergence rate measured by the residual between two successive iterates, where N represents the iteration counter. In some cases, considering that the underlying subproblems of PPD+ cannot be easily solved exactly or up to a high precision, we further propose two inexact versions of the PPD+ under absolute and relative error criteria. Finally, we employ a restarting technique to enhance our algorithms for the purpose of making them more robust and efficient. A series of numerical experiments demonstrate that our algorithms perform well in practice.
AbstractList	In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Proximal Primal-Dual algorithm (PPD+), which possesses an O ( 1 / N 2 ) convergence rate measured by the residual between two successive iterates, where N represents the iteration counter. In some cases, considering that the underlying subproblems of PPD+ cannot be easily solved exactly or up to a high precision, we further propose two inexact versions of the PPD+ under absolute and relative error criteria. Finally, we employ a restarting technique to enhance our algorithms for the purpose of making them more robust and efficient. A series of numerical experiments demonstrate that our algorithms perform well in practice. In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Proximal Primal-Dual algorithm (PPD+), which possesses an O(1/N2) convergence rate measured by the residual between two successive iterates, where N represents the iteration counter. In some cases, considering that the underlying subproblems of PPD+ cannot be easily solved exactly or up to a high precision, we further propose two inexact versions of the PPD+ under absolute and relative error criteria. Finally, we employ a restarting technique to enhance our algorithms for the purpose of making them more robust and efficient. A series of numerical experiments demonstrate that our algorithms perform well in practice.
Author	Qu, Yunfei He, Hongjin Han, Deren Zhang, Tao
Author_xml	– sequence: 1 givenname: Yunfei surname: Qu fullname: Qu, Yunfei organization: School of Science, China University of Geosciences – sequence: 2 givenname: Hongjin surname: He fullname: He, Hongjin organization: School of Mathematics and Statistics, Ningbo University – sequence: 3 givenname: Tao surname: Zhang fullname: Zhang, Tao organization: School of Mathematics and Statistics, Linyi University – sequence: 4 givenname: Deren orcidid: 0000-0002-0427-8224 surname: Han fullname: Han, Deren email: handr@buaa.edu.cn organization: LMIB, School of Mathematical Sciences, Beihang University
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Cites_doi	10.1137/140963467 10.1137/20M1379344 10.1007/s10589-020-00186-y 10.1137/21M1408683 10.1007/s10915-025-02802-7 10.1137/18M1224568 10.1007/s12190-023-01899-z 10.1007/s10589-017-9911-z 10.1007/s10107-022-01901-9 10.1109/ICASSP40776.2020.9053722 10.1007/s10851-010-0251-1 10.1109/TMI.2015.2437894 10.1137/10081602X 10.1007/s11590-021-01775-4 10.1137/19M1293855 10.1137/21M1395685 10.1016/0041-5553(85)90100-4 10.1007/s11228-017-0421-z 10.1137/1.9781611973402.110 10.1007/s11075-021-01069-x 10.1016/j.orl.2015.02.001 10.1007/s10957-020-01806-7 10.1111/j.1467-9868.2005.00490.x 10.1007/s00211-013-0580-2 10.1137/110836936 10.1016/j.orl.2024.107199 10.1090/mcom/3610 10.1137/09076934X 10.1137/21M1453463 10.1007/s10208-013-9150-3 10.1007/s11075-023-01509-w 10.1287/moor.27.4.681.305 10.1007/s40305-024-00568-7 10.1088/0266-5611/29/2/025011 10.1007/s10589-017-9909-6 10.1137/100814494 10.1007/s10915-023-02105-9 10.1007/s10107-012-0629-5 10.1093/imanum/drz007 10.1109/CDC.2014.7040179 10.1007/s10957-012-0245-9 10.1016/j.disopt.2017.08.004 10.1137/18M1213488 10.1137/16M1093094 10.1007/s10915-020-01330-w 10.1186/s13663-016-0543-2 10.1007/s00211-014-0673-6 10.1137/18M1182851 10.1007/s10915-018-0680-3 10.1007/s10444-011-9254-8 10.1007/s10107-015-0957-3 10.1007/s10107-017-1164-1
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References	E Esser (1545_CR16) 2010; 3 J Liang (1545_CR35) 2022; 44 1545_CR20 R Tibshirani (1545_CR47) 2005; 67 AS Nemirovskii (1545_CR38) 1985; 25 1545_CR60 P Chen (1545_CR7) 2013; 29 F Jiang (1545_CR29) 2023; 94 L Condat (1545_CR9) 2013; 158 1545_CR2 A d’Aspremont (1545_CR11) 2021; 5 BC Vũ (1545_CR53) 2013; 38 Q Tran-Dinh (1545_CR51) 2020; 30 O Fercoq (1545_CR17) 2019; 39 F Jiang (1545_CR31) 2021; 88 1545_CR36 P Le Bodic (1545_CR34) 2018; 27 J Eckstein (1545_CR15) 2017; 68 1545_CR32 L Condat (1545_CR10) 2023; 65 D Davis (1545_CR12) 2017; 25 T Yang (1545_CR57) 2018; 19 V Roulet (1545_CR43) 2020; 30 T Valkonen (1545_CR52) 2020; 30 D Applegate (1545_CR1) 2023; 201 M Yan (1545_CR56) 2018; 76 F Jiang (1545_CR30) 2021; 90 Y Nesterov (1545_CR39) 2013; 140 B He (1545_CR23) 2022; 15 B He (1545_CR26) 2015; 130 F Shi (1545_CR45) 2015; 34 Y Zhu (1545_CR61) 2022; 32 P Chen (1545_CR8) 2016; 2016 H Oliver (1545_CR41) 2024; 57 Y Drori (1545_CR14) 2015; 43 Q Tran-Dinh (1545_CR49) 2022; 16 Y Ma (1545_CR37) 2023; 94 A Chambolle (1545_CR5) 2011; 40 M Ito (1545_CR28) 2021; 188 A Chambolle (1545_CR6) 2016; 159 B He (1545_CR25) 2012; 50 J Rasch (1545_CR42) 2020; 76 J Bai (1545_CR3) 2025; 102 LM Briceno-Arias (1545_CR4) 2011; 21 B He (1545_CR24) 2012; 5 B O’donoghue (1545_CR40) 2015; 15 AS Schulz (1545_CR44) 2002; 27 A Torrico (1545_CR48) 2021; 33 K Wang (1545_CR55) 2023; 69 BS He (1545_CR27) 2015; 7 1545_CR58 1545_CR59 D Han (1545_CR22) 2014; 127 1545_CR13 Q Tran-Dinh (1545_CR50) 2018; 28 P Latafat (1545_CR33) 2017; 68 K Wang (1545_CR54) 2020; 85 1545_CR19 RM Freund (1545_CR18) 2018; 170 W Su (1545_CR46) 2016; 17 EY Hamedani (1545_CR21) 2021; 31
References_xml	– volume: 7 start-page: 2526 year: 2015 ident: 1545_CR27 publication-title: SIAM J. Imaging Sci. doi: 10.1137/140963467 – volume: 65 start-page: 375 issue: 2 year: 2023 ident: 1545_CR10 publication-title: SIAM Rev. doi: 10.1137/20M1379344 – volume: 76 start-page: 381 issue: 2 year: 2020 ident: 1545_CR42 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-020-00186-y – volume: 32 start-page: 2580 issue: 4 year: 2022 ident: 1545_CR61 publication-title: SIAM J. Optim. doi: 10.1137/21M1408683 – volume: 102 start-page: 80 issue: 3 year: 2025 ident: 1545_CR3 publication-title: J. Sci. Comput. doi: 10.1007/s10915-025-02802-7 – volume: 30 start-page: 262 issue: 1 year: 2020 ident: 1545_CR43 publication-title: SIAM J. Optim. doi: 10.1137/18M1224568 – volume: 69 start-page: 3693 issue: 5 year: 2023 ident: 1545_CR55 publication-title: J. Appl. Math. Comput. doi: 10.1007/s12190-023-01899-z – volume: 68 start-page: 363 issue: 2 year: 2017 ident: 1545_CR15 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-017-9911-z – volume: 201 start-page: 133 issue: 1–2 year: 2023 ident: 1545_CR1 publication-title: Math. Program. doi: 10.1007/s10107-022-01901-9 – ident: 1545_CR59 – ident: 1545_CR60 – ident: 1545_CR13 doi: 10.1109/ICASSP40776.2020.9053722 – ident: 1545_CR36 – volume: 40 start-page: 120 year: 2011 ident: 1545_CR5 publication-title: J. Math. Imaging Vis. doi: 10.1007/s10851-010-0251-1 – ident: 1545_CR32 – volume: 34 start-page: 2459 issue: 12 year: 2015 ident: 1545_CR45 publication-title: IEEE Trans. Med. Imaging doi: 10.1109/TMI.2015.2437894 – volume: 33 start-page: 1590 issue: 4 year: 2021 ident: 1545_CR48 publication-title: INFORMS J. Comput. – volume: 19 start-page: 236 issue: 1 year: 2018 ident: 1545_CR57 publication-title: J. Mach. Learn. Res. – volume: 21 start-page: 1230 issue: 4 year: 2011 ident: 1545_CR4 publication-title: SIAM J. Optim. doi: 10.1137/10081602X – volume: 16 start-page: 1235 issue: 4 year: 2022 ident: 1545_CR49 publication-title: Optim. Lett. doi: 10.1007/s11590-021-01775-4 – volume: 30 start-page: 2866 issue: 4 year: 2020 ident: 1545_CR51 publication-title: SIAM J. Optim. doi: 10.1137/19M1293855 – volume: 44 start-page: A1069 issue: 3 year: 2022 ident: 1545_CR35 publication-title: SIAM J. Sci. Comput. doi: 10.1137/21M1395685 – volume: 25 start-page: 21 issue: 2 year: 1985 ident: 1545_CR38 publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(85)90100-4 – volume: 25 start-page: 829 year: 2017 ident: 1545_CR12 publication-title: Set-Valued Var. Anal. doi: 10.1007/s11228-017-0421-z – ident: 1545_CR2 doi: 10.1137/1.9781611973402.110 – volume: 88 start-page: 1109 year: 2021 ident: 1545_CR31 publication-title: Numer. Algorithms doi: 10.1007/s11075-021-01069-x – volume: 43 start-page: 209 issue: 2 year: 2015 ident: 1545_CR14 publication-title: Oper. Res. Lett. doi: 10.1016/j.orl.2015.02.001 – volume: 188 start-page: 770 year: 2021 ident: 1545_CR28 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-020-01806-7 – volume: 67 start-page: 91 year: 2005 ident: 1545_CR47 publication-title: J. Roy. Statist. Soc. Ser. B doi: 10.1111/j.1467-9868.2005.00490.x – volume: 127 start-page: 167 year: 2014 ident: 1545_CR22 publication-title: Numer. Math. doi: 10.1007/s00211-013-0580-2 – ident: 1545_CR20 – volume: 50 start-page: 700 year: 2012 ident: 1545_CR25 publication-title: SIAM J. Numer. Anal. doi: 10.1137/110836936 – volume: 57 year: 2024 ident: 1545_CR41 publication-title: Oper. Res. Lett. doi: 10.1016/j.orl.2024.107199 – volume: 17 start-page: 1 year: 2016 ident: 1545_CR46 publication-title: J. Mach. Learn. Res. – volume: 90 start-page: 1227 issue: 329 year: 2021 ident: 1545_CR30 publication-title: Math. Comput. doi: 10.1090/mcom/3610 – volume: 3 start-page: 1015 year: 2010 ident: 1545_CR16 publication-title: SIAM J. Imaging Sci. doi: 10.1137/09076934X – volume: 15 start-page: 1157 issue: 3 year: 2022 ident: 1545_CR23 publication-title: SIAM J. Imaging Sci. doi: 10.1137/21M1453463 – volume: 15 start-page: 715 year: 2015 ident: 1545_CR40 publication-title: Found. Comput. Math. doi: 10.1007/s10208-013-9150-3 – volume: 94 start-page: 479 year: 2023 ident: 1545_CR29 publication-title: Numer. Algorithms doi: 10.1007/s11075-023-01509-w – volume: 27 start-page: 681 issue: 4 year: 2002 ident: 1545_CR44 publication-title: Math. Oper. Res. doi: 10.1287/moor.27.4.681.305 – ident: 1545_CR58 doi: 10.1007/s40305-024-00568-7 – volume: 29 issue: 2 year: 2013 ident: 1545_CR7 publication-title: Inverse Probl. doi: 10.1088/0266-5611/29/2/025011 – volume: 68 start-page: 57 year: 2017 ident: 1545_CR33 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-017-9909-6 – volume: 5 start-page: 119 year: 2012 ident: 1545_CR24 publication-title: SIAM J. Imaging Sci. doi: 10.1137/100814494 – volume: 94 start-page: 60 issue: 3 year: 2023 ident: 1545_CR37 publication-title: J. Sci. Comput. doi: 10.1007/s10915-023-02105-9 – volume: 140 start-page: 125 issue: 1 year: 2013 ident: 1545_CR39 publication-title: Math. Program. doi: 10.1007/s10107-012-0629-5 – volume: 5 start-page: 1 issue: 1–2 year: 2021 ident: 1545_CR11 publication-title: Acceleration methods. Found. Trends Optim. – volume: 39 start-page: 2069 issue: 4 year: 2019 ident: 1545_CR17 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/drz007 – ident: 1545_CR19 doi: 10.1109/CDC.2014.7040179 – volume: 158 start-page: 460 issue: 2 year: 2013 ident: 1545_CR9 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-012-0245-9 – volume: 27 start-page: 1 year: 2018 ident: 1545_CR34 publication-title: Discrete Optim. doi: 10.1016/j.disopt.2017.08.004 – volume: 31 start-page: 1299 issue: 2 year: 2021 ident: 1545_CR21 publication-title: SIAM J. Optim. doi: 10.1137/18M1213488 – volume: 28 start-page: 96 issue: 1 year: 2018 ident: 1545_CR50 publication-title: SIAM J. Optim. doi: 10.1137/16M1093094 – volume: 85 start-page: 30 issue: 2 year: 2020 ident: 1545_CR54 publication-title: J. Sci. Comput. doi: 10.1007/s10915-020-01330-w – volume: 2016 start-page: 1 issue: 1 year: 2016 ident: 1545_CR8 publication-title: Fixed Point Theory Appl. doi: 10.1186/s13663-016-0543-2 – volume: 130 start-page: 567 issue: 3 year: 2015 ident: 1545_CR26 publication-title: Numer. Math. doi: 10.1007/s00211-014-0673-6 – volume: 30 start-page: 1391 issue: 2 year: 2020 ident: 1545_CR52 publication-title: SIAM J. Optim. doi: 10.1137/18M1182851 – volume: 76 start-page: 1698 year: 2018 ident: 1545_CR56 publication-title: J. Sci. Comput. doi: 10.1007/s10915-018-0680-3 – volume: 38 start-page: 667 issue: 3 year: 2013 ident: 1545_CR53 publication-title: Adv. Comput. Math. doi: 10.1007/s10444-011-9254-8 – volume: 159 start-page: 253 issue: 1–2 year: 2016 ident: 1545_CR6 publication-title: Math. Program. doi: 10.1007/s10107-015-0957-3 – volume: 170 start-page: 445 year: 2018 ident: 1545_CR18 publication-title: Math. Program. doi: 10.1007/s10107-017-1164-1
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SubjectTerms	Algorithms Computer Science Convex analysis Euclidean space Lipschitz condition Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Saddle points
Subtitle	Practical proximal primal-dual algorithms for structured saddle point problems
Title	Practical proximal primal-dual algorithms for structured saddle point problems
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