{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T22:51:51Z","timestamp":1771800711568,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T00:00:00Z","timestamp":1642032000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T00:00:00Z","timestamp":1642032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["11871279"],"award-info":[{"award-number":["11871279"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["11971238"],"award-info":[{"award-number":["11971238"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["12001286"],"award-info":[{"award-number":["12001286"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["11571178"],"award-info":[{"award-number":["11571178"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["11871279"],"award-info":[{"award-number":["11871279"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"usa national science foundation","doi-asserted-by":"crossref","award":["1819161"],"award-info":[{"award-number":["1819161"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000001","name":"usa national science foundation","doi-asserted-by":"crossref","award":["2110722"],"award-info":[{"award-number":["2110722"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2022,7]]},"DOI":"10.1007\/s11590-021-01832-y","type":"journal-article","created":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T00:04:26Z","timestamp":1642032266000},"page":"1675-1700","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Unified linear convergence of first-order primal-dual algorithms for saddle point problems"],"prefix":"10.1007","volume":"16","author":[{"given":"Fan","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Zhongming","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Xingju","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Hongchao","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,13]]},"reference":[{"key":"1832_CR1","doi-asserted-by":"publisher","first-page":"1419","DOI":"10.1007\/s10898-012-9999-8","volume":"57","author":"X Cai","year":"2013","unstructured":"Cai, X., Han, D., Xu, L.: An improved first-order primal-dual algorithm with a new correction step. J. Global Optim. 57, 1419\u20131428 (2013)","journal-title":"J. Global Optim."},{"key":"1832_CR2","doi-asserted-by":"publisher","first-page":"2783","DOI":"10.1137\/17M1134834","volume":"28","author":"A Chambolle","year":"2018","unstructured":"Chambolle, A., Ehrhardt, M.J., Richt\u00e1rik, P., Schonlieb, C.-B.: Stochastic primal-dual hybrid gradient algorithm with arbitrary sampling and imaging applications. SIAM J. Optim. 28, 2783\u20132808 (2018)","journal-title":"SIAM J. Optim."},{"key":"1832_CR3","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","volume":"40","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120\u2013145 (2011)","journal-title":"J. Math. Imaging Vis."},{"key":"1832_CR4","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s10107-015-0957-3","volume":"159","author":"A Chambolle","year":"2016","unstructured":"Chambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal-dual algorithm. Math. Program. 159, 253\u2013287 (2016)","journal-title":"Math. Program."},{"key":"1832_CR5","doi-asserted-by":"publisher","first-page":"025011","DOI":"10.1088\/0266-5611\/29\/2\/025011","volume":"29","author":"P Chen","year":"2013","unstructured":"Chen, P., Huang, J., Zhang, X.: A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Prob. 29, 025011 (2013)","journal-title":"Inverse Prob."},{"key":"1832_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13663-016-0543-2","volume":"2016","author":"P Chen","year":"2016","unstructured":"Chen, P., Huang, J., Zhang, X.: A primal-dual fixed point algorithm for minimization of the sum of three convex separable functions. Fixed Point Theory Appl. 2016, 1\u201318 (2016)","journal-title":"Fixed Point Theory Appl."},{"key":"1832_CR7","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1007\/s10957-012-0245-9","volume":"158","author":"L Condat","year":"2013","unstructured":"Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158, 460\u2013479 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1832_CR8","series-title":"Springer Monographs in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-87821-8","volume-title":"Implicit Functions and Solution Mappings","author":"AL Dontchev","year":"2009","unstructured":"Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer Monographs in Mathematics, vol. 208. Springer, Berlin (2009)"},{"key":"1832_CR9","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.orl.2015.02.001","volume":"43","author":"Y Drori","year":"2015","unstructured":"Drori, Y., Sabach, S., Teboulle, M.: A simple algorithm for a class of nonsmooth convex-concave saddle-point problems. Oper. Res. Lett. 43, 209\u2013214 (2015)","journal-title":"Oper. Res. Lett."},{"key":"1832_CR10","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1137\/09076934X","volume":"3","author":"E Esser","year":"2010","unstructured":"Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imag. Sci. 3, 1015\u20131046 (2010)","journal-title":"SIAM J. Imag. Sci."},{"key":"1832_CR11","doi-asserted-by":"crossref","unstructured":"Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, pp. 299\u2013331. North-Holland, Amsterdam (1983)","DOI":"10.1016\/S0168-2024(08)70034-1"},{"key":"1832_CR12","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0898-1221(76)90003-1","volume":"2","author":"D Gabay","year":"1976","unstructured":"Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2, 17\u201340 (1976)","journal-title":"Comput. Math. Appl."},{"key":"1832_CR13","doi-asserted-by":"crossref","unstructured":"Glowinski, R., Marroco, A.: Sur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9 d\u2019une classe de probl\u00e8mes de dirichlet non lin\u00e9aires, ESAIM: Mathematical Modelling and Numerical Analysis-Mod\u00e9lisation Math\u00e9matique et Analyse Num\u00e9rique, 9 (1975), pp.\u00a041\u201376","DOI":"10.1051\/m2an\/197509R200411"},{"key":"1832_CR14","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s00211-013-0580-2","volume":"127","author":"D Han","year":"2014","unstructured":"Han, D., He, H., Yang, H., Yuan, X.: A customized Douglas\u2013Rachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167\u2013200 (2014)","journal-title":"Numer. Math."},{"key":"1832_CR15","doi-asserted-by":"publisher","first-page":"622","DOI":"10.1287\/moor.2017.0875","volume":"43","author":"D Han","year":"2017","unstructured":"Han, D., Sun, D., Zhang, L.: Linear rate convergence of the alternating direction method of multipliers for convex composite programming. Math. Oper. Res. 43, 622\u2013637 (2017)","journal-title":"Math. Oper. Res."},{"key":"1832_CR16","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1137\/100814494","volume":"5","author":"B He","year":"2012","unstructured":"He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imag. Sci. 5, 119\u2013149 (2012)","journal-title":"SIAM J. Imag. Sci."},{"key":"1832_CR17","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s10898-016-0437-1","volume":"66","author":"H He","year":"2016","unstructured":"He, H., Desai, J., Wang, K.: A primal-dual prediction-correction algorithm for saddle point optimization. J. Global Optim. 66, 573\u2013583 (2016)","journal-title":"J. Global Optim."},{"key":"1832_CR18","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1090\/mcom\/3610","volume":"90","author":"F Jiang","year":"2021","unstructured":"Jiang, F., Cai, X., Wu, Z., Han, D.: Approximate first-order primal-dual algorithms for saddle point problems. Math. Comput. 90, 1227\u20131262 (2021)","journal-title":"Math. Comput."},{"key":"1832_CR19","doi-asserted-by":"crossref","unstructured":"Jiang, F., Wu, Z., Cai, X., Zhang, H.: A first-order inexact primal-dual algorithm for a class of convex-concave saddle point problems. Numer. Algorithms 88, 1109\u20131136 (2021)","DOI":"10.1007\/s11075-021-01069-x"},{"key":"1832_CR20","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10589-017-9909-6","volume":"68","author":"P Latafat","year":"2017","unstructured":"Latafat, P., Patrinos, P.: Asymmetric forward-backward-adjoint splitting for solving monotone inclusions involving three operators. Comput. Optim. Appl. 68, 57\u201393 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"1832_CR21","doi-asserted-by":"crossref","unstructured":"Liu, J., Yuan, L., Ye, J.: An efficient algorithm for a class of fused lasso problems. In: Proceedings of the 16th ACM SIGKDD International Donference on Knowledge Discovery and Data Mining, ACM, pp.\u00a0323\u2013332 (2010)","DOI":"10.1145\/1835804.1835847"},{"key":"1832_CR22","doi-asserted-by":"publisher","first-page":"125007","DOI":"10.1088\/0266-5611\/27\/12\/125007","volume":"27","author":"I Loris","year":"2011","unstructured":"Loris, I., Verhoeven, C.: On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty. Inverse Prob. 27, 125007 (2011)","journal-title":"Inverse Prob."},{"key":"1832_CR23","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1137\/16M1092015","volume":"28","author":"Y Malitsky","year":"2018","unstructured":"Malitsky, Y., Pock, T.: A first-order primal-dual algorithm with linesearch. SIAM J. Optim. 28, 411\u2013432 (2018)","journal-title":"SIAM J. Optim."},{"key":"1832_CR24","unstructured":"Mokhtari, A., Ozdaglar, A., Pattathil, S.: A unified analysis of extra-gradient and optimistic gradient methods for saddle point problems: Proximal point approach. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp.\u00a01497\u20131507 (2020)"},{"key":"1832_CR25","doi-asserted-by":"publisher","first-page":"3230","DOI":"10.1137\/19M127375X","volume":"30","author":"A Mokhtari","year":"2020","unstructured":"Mokhtari, A., Ozdaglar, A.E., Pattathil, S.: Convergence rate of O(1\/k) for optimistic gradient and extragradient methods in smooth convex-concave saddle point problems. SIAM J. Optim. 30, 3230\u20133251 (2020)","journal-title":"SIAM J. Optim."},{"key":"1832_CR26","doi-asserted-by":"publisher","first-page":"2200","DOI":"10.1016\/j.cam.2009.10.006","volume":"233","author":"B Morini","year":"2010","unstructured":"Morini, B., Porcelli, M., Chan, R.H.: A reduced Newton method for constrained linear least-squares problems. J. Comput. Appl. Math. 233, 2200\u20132212 (2010)","journal-title":"J. Comput. Appl. Math."},{"key":"1832_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-91578-4","volume-title":"Lectures on Convex Optimization","author":"Y Nesterov","year":"2018","unstructured":"Nesterov, Y.: Lectures on Convex Optimization, vol. 137. Springer, Berlin (2018)"},{"key":"1832_CR28","doi-asserted-by":"crossref","unstructured":"O\u2019Connor, D., Vandenberghe, L.: On the equivalence of the primal-dual hybrid gradient method and Douglas\u2013Rachford splitting. Math. Program. 179, 85\u2013108 (2020)","DOI":"10.1007\/s10107-018-1321-1"},{"key":"1832_CR29","first-page":"127","volume":"1","author":"N Parikh","year":"2014","unstructured":"Parikh, N., Boyd, S., et al.: Proximal algorithms, Foundations and Trends\u00ae. Optimization 1, 127\u2013239 (2014)","journal-title":"Optimization"},{"key":"1832_CR30","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s10589-020-00186-y","volume":"2","author":"J Rasch","year":"2020","unstructured":"Rasch, J., Chambolle, A.: Inexact first-order primal-dual algorithms. Comput. Optim. Appl. 2, 381\u2013430 (2020)","journal-title":"Comput. Optim. Appl."},{"key":"1832_CR31","unstructured":"Robinson, S.M.: An implicit-function theorem for generalized variational inequalities. tech. rep., Wisconsin Univ Madison Mathmatics Research Center, (1976)"},{"key":"1832_CR32","doi-asserted-by":"crossref","unstructured":"Robinson, S.M.: Some continuity properties of polyhedral multifunctions. In: K\u00f6nig, H., Korte, B., Ritter, K. (eds.) Mathematical Programming at Oberwolfach, pp. 206\u2013214. Springer, Berlin (1981)","DOI":"10.1007\/BFb0120929"},{"key":"1832_CR33","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer Science & Business Media, Berlin (2009)"},{"key":"1832_CR34","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","volume":"60","author":"LI Rudin","year":"1992","unstructured":"Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259\u2013268 (1992)","journal-title":"Phys. D"},{"key":"1832_CR35","doi-asserted-by":"crossref","unstructured":"Shi, F., Cheng, J., Wang, L., Yap, P.-T., Shen, D.: Low-rank total variation for image super-resolution. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, pp.\u00a0155\u2013162 (2013)","DOI":"10.1007\/978-3-642-40811-3_20"},{"key":"1832_CR36","doi-asserted-by":"publisher","first-page":"2459","DOI":"10.1109\/TMI.2015.2437894","volume":"34","author":"F Shi","year":"2015","unstructured":"Shi, F., Cheng, J., Wang, L., Yap, P.-T., Shen, D.: LRTV: MR image super-resolution with low-rank and total variation regularizations. IEEE Trans. Med. Imaging 34, 2459\u20132466 (2015)","journal-title":"IEEE Trans. Med. Imaging"},{"key":"1832_CR37","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1111\/j.1467-9868.2005.00490.x","volume":"67","author":"R Tibshirani","year":"2005","unstructured":"Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., Knight, K.: Sparsity and smoothness via the fused lasso. J. Royal Stat. Soc. Ser. B (Stat. Methodol.) 67, 91\u2013108 (2005)","journal-title":"J. Royal Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"1832_CR38","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1007\/s10444-011-9254-8","volume":"38","author":"BC V\u0169","year":"2013","unstructured":"V\u0169, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667\u2013681 (2013)","journal-title":"Adv. Comput. Math."},{"key":"1832_CR39","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1007\/s10915-018-0680-3","volume":"76","author":"M Yan","year":"2018","unstructured":"Yan, M.: A new primal-dual algorithm for minimizing the sum of three functions with a linear operator. J. Sci. Comput. 76, 1698\u20131717 (2018)","journal-title":"J. Sci. Comput."},{"key":"1832_CR40","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","volume":"68","author":"M Yuan","year":"2006","unstructured":"Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. Royal Stat. Soc. Ser. B (Statistical Methodology) 68, 49\u201367 (2006)","journal-title":"J. Royal Stat. Soc. Ser. B (Statistical Methodology)"},{"key":"1832_CR41","unstructured":"Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 34 (2008)"}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-021-01832-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-021-01832-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-021-01832-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,6]],"date-time":"2022-06-06T11:19:48Z","timestamp":1654514388000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-021-01832-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,13]]},"references-count":41,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["1832"],"URL":"https:\/\/doi.org\/10.1007\/s11590-021-01832-y","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,13]]},"assertion":[{"value":"22 June 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}