{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T02:52:14Z","timestamp":1776394334028,"version":"3.51.2"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"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":["11871029"],"award-info":[{"award-number":["11871029"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["12302318"],"award-info":[{"award-number":["12302318"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2022,7]]},"DOI":"10.1007\/s10851-022-01089-9","type":"journal-article","created":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T21:02:53Z","timestamp":1651266173000},"page":"662-671","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On Convergence of the Arrow\u2013Hurwicz Method for Saddle Point Problems"],"prefix":"10.1007","volume":"64","author":[{"given":"Bingsheng","family":"He","sequence":"first","affiliation":[]},{"given":"Shengjie","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Xiaoming","family":"Yuan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"key":"1089_CR1","volume-title":"Studies in Linear and Non-linear Programming","author":"KJ Arrow","year":"1958","unstructured":"Arrow, K.J., Hurwicz, L., Uzawa, H.: Studies in Linear and Non-linear Programming. Stanford University Press, Stanford (1958)"},{"key":"1089_CR2","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1137\/050630714","volume":"44","author":"C Bacuta","year":"2006","unstructured":"Bacuta, C.: A unified approach for Uzawa algorithms. SIAM J. Numer. Anal. 44, 2633\u20132649 (2006)","journal-title":"SIAM J. Numer. Anal."},{"key":"1089_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492904000212","volume":"14","author":"M Benzi","year":"2005","unstructured":"Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1\u2013137 (2005)","journal-title":"Acta Numer."},{"key":"1089_CR4","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s10851-011-0324-9","volume":"44","author":"S Bonettini","year":"2012","unstructured":"Bonettini, S., Ruggiero, V.: On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration. J. Math. Imaging Vis. 44, 236\u2013253 (2012)","journal-title":"J. Math. Imaging Vis."},{"key":"1089_CR5","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1137\/S0036142994273343","volume":"34","author":"JH Bramble","year":"1997","unstructured":"Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072\u20131092 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"1089_CR6","volume-title":"Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics","author":"F Brezzi","year":"2012","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (2012)"},{"key":"1089_CR7","doi-asserted-by":"crossref","unstructured":"Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery, Volume 9 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter, Berlin, pp. 263\u2013340 (2010)","DOI":"10.1515\/9783110226157.263"},{"key":"1089_CR8","doi-asserted-by":"crossref","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\u2013dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120\u2013145 (2011)","journal-title":"J. Math. Imaging Vis."},{"key":"1089_CR9","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1017\/S096249291600009X","volume":"25","author":"A Chambolle","year":"2016","unstructured":"Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161\u2013319 (2016)","journal-title":"Acta Numer."},{"key":"1089_CR10","doi-asserted-by":"crossref","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\u2013dual algorithm. Math. Program. 159, 253\u2013287 (2016)","journal-title":"Math. Program."},{"key":"1089_CR11","doi-asserted-by":"crossref","DOI":"10.1017\/9781139171984","volume-title":"Introduction to Numerical Linear Algebra and Optimization","author":"PG Ciarlet","year":"1989","unstructured":"Ciarlet, P.G., Miara, B., Thomas, J.-M.: Introduction to Numerical Linear Algebra and Optimization. Cambridge University Press, Cambridge (1989)"},{"key":"1089_CR12","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1137\/0731085","volume":"31","author":"HC Elman","year":"1994","unstructured":"Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645\u20131661 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"1089_CR13","doi-asserted-by":"crossref","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\u2013dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3, 1015\u20131046 (2010)","journal-title":"SIAM J. Imaging Sci."},{"key":"1089_CR14","unstructured":"Goldstein, T., Li, M., Yuan, X.M.: Adaptive primal-dual splitting methods for statistical learning and image processing. In: Advances in Neural Information Processing Systems, pp. 2089\u20132097 (2015)"},{"key":"1089_CR15","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s10851-017-0709-5","volume":"58","author":"BS He","year":"2017","unstructured":"He, B.S., Ma, F., Yuan, X.M.: An algorithmic framework of generalized primal\u2013dual hybrid gradient methods for saddle point problems. J. Math. Imaging Vis. 58, 279\u2013293 (2017)","journal-title":"J. Math. Imaging Vis."},{"key":"1089_CR16","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1137\/140963467","volume":"7","author":"BS He","year":"2014","unstructured":"He, B.S., You, Y.F., Yuan, X.M.: On the convergence of primal\u2013dual hybrid gradient algorithm. SIAM J. Imaging Sci. 7, 2526\u20132537 (2014)","journal-title":"SIAM J. Imaging Sci."},{"key":"1089_CR17","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1137\/100814494","volume":"5","author":"BS He","year":"2012","unstructured":"He, B.S., Yuan, X.M.: Convergence analysis of primal\u2013dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119\u2013149 (2012)","journal-title":"SIAM J. Imaging Sci."},{"key":"1089_CR18","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF00927673","volume":"4","author":"MR Hestenes","year":"1969","unstructured":"Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303\u2013320 (1969)","journal-title":"J. Optim. Theory Appl."},{"key":"1089_CR19","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1137\/S1052623403428683","volume":"16","author":"Q Hu","year":"2006","unstructured":"Hu, Q., Zou, J.: Nonlinear inexact Uzawa algorithms for linear and nonlinear saddle-point problems. SIAM J. Optim. 16, 798\u2013825 (2006)","journal-title":"SIAM J. Optim."},{"key":"1089_CR20","volume-title":"Numerical Analysis: Mathematics of Scientific Computing","author":"D Kincaid","year":"2002","unstructured":"Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. American Mathematical Society, Providence (2002)","edition":"3"},{"issue":"4","key":"1089_CR21","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody. 12(4), 747\u2013756 (1976)","journal-title":"Ekon. Mat. Metody."},{"key":"1089_CR22","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-74503-9","volume-title":"Linear and Nonlinear Programming","author":"DG Luenberger","year":"2008","unstructured":"Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming. Springer, New York (2008)"},{"key":"1089_CR23","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1137\/16M1092015","volume":"28","author":"Y Malitsky","year":"2018","unstructured":"Malitsky, Y., Pock, T.: A first-order primal\u2013dual algorithm with line search. SIAM J. Optim. 28, 411\u2013432 (2018)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1089_CR24","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1137\/140976601","volume":"8","author":"T M\u00f6llenhoff","year":"2015","unstructured":"M\u00f6llenhoff, T., Strekalovskiy, E., Moeller, M., Cremers, D.: The primal\u2013dual hybrid gradient method for semiconvex splittings. SIAM J. Imaging Sci. 8(2), 827\u2013857 (2015)","journal-title":"SIAM J. Imaging Sci."},{"issue":"1","key":"1089_CR25","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1137\/130934921","volume":"57","author":"J Pestana","year":"2015","unstructured":"Pestana, J., Wathen, A.J.: Natural preconditioning and iterative methods for saddle point systems. SIAM Rev. 57(1), 71\u201391 (2015)","journal-title":"SIAM Rev."},{"key":"1089_CR26","doi-asserted-by":"crossref","unstructured":"Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal\u2013dual algorithms in convex optimization. In: IEEE International Conference on Computer Vision, pp. 1762\u20131769 (2011)","DOI":"10.1109\/ICCV.2011.6126441"},{"issue":"5","key":"1089_CR27","first-page":"777","volume":"28","author":"LD Popov","year":"1980","unstructured":"Popov, L.D.: A modification of the Arrow\u2013Hurwitz method of search for saddle points. Mat. Zametki. 28(5), 777\u2013784 (1980)","journal-title":"Mat. Zametki."},{"key":"1089_CR28","volume-title":"Optimization","author":"MJ Powell","year":"1969","unstructured":"Powell, M.J.: A method for non-linear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization. Academic Press, New York (1969)"},{"key":"1089_CR29","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics","author":"A Quarteroni","year":"1994","unstructured":"Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)"},{"key":"1089_CR30","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1137\/0726057","volume":"26","author":"W Queck","year":"1989","unstructured":"Queck, W.: The convergence factor of preconditioned algorithms of the Arrow\u2013Hurwicz type. SIAM J. Numer. Anal. 26, 1016\u20131030 (1989)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1\u20134","key":"1089_CR31","doi-asserted-by":"crossref","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. Physica D 60(1\u20134), 259\u2013268 (1992)","journal-title":"Physica D"},{"key":"1089_CR32","doi-asserted-by":"crossref","first-page":"2656","DOI":"10.1137\/19M1245736","volume":"57","author":"Y Song","year":"2019","unstructured":"Song, Y., Yuan, X., Yue, H.: An inexact Uzawa algorithmic framework for nonlinear saddle point problems with applications to elliptic optimal control problem. SIAM J. Numer. Anal. 57, 2656\u20132684 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1089_CR33","volume-title":"Introduction to Numerical Analysis. Texts in Applied Mathematics","author":"J Stoer","year":"2013","unstructured":"Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Texts in Applied Mathematics, vol. 12. Springer, New York (2013)"},{"key":"1089_CR34","volume-title":"Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics","author":"P Wesseling","year":"2001","unstructured":"Wesseling, P.: Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics, vol. 29. Springer, Berlin (2001)"},{"key":"1089_CR35","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s10915-010-9408-8","volume":"46","author":"X Zhang","year":"2011","unstructured":"Zhang, X., Burger, M., Osher, S.: A unified primal\u2013dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20\u201346 (2011)","journal-title":"J. Sci. Comput."},{"key":"1089_CR36","unstructured":"Zhu, M., Chan, T.: An efficient primal\u2013dual hybrid gradient algorithm for total variation image restoration. CAM report 08\u201334, UCLA, Los Angeles, CA (2008)"}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-022-01089-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10851-022-01089-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-022-01089-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T21:46:25Z","timestamp":1700516785000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10851-022-01089-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,29]]},"references-count":36,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["1089"],"URL":"https:\/\/doi.org\/10.1007\/s10851-022-01089-9","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"value":"0924-9907","type":"print"},{"value":"1573-7683","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,29]]},"assertion":[{"value":"12 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 March 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}