{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:54:41Z","timestamp":1768294481888,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T00:00:00Z","timestamp":1698883200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T00:00:00Z","timestamp":1698883200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000995","name":"Australian National University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000995","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we consider from two different aspects the proximal alternating direction method of multipliers (ADMM) in Hilbert spaces. We first consider the application of the proximal ADMM to solve well-posed linearly constrained two-block separable convex minimization problems in Hilbert spaces and obtain new and improved non-ergodic convergence rate results, including linear and sublinear rates under certain regularity conditions. We next consider the proximal ADMM as a regularization method for solving linear ill-posed inverse problems in Hilbert spaces. When the data is corrupted by additive noise, we establish, under a benchmark source condition, a convergence rate result in terms of the noise level when the number of iterations is properly chosen.<\/jats:p>","DOI":"10.1007\/s10915-023-02383-3","type":"journal-article","created":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T07:02:05Z","timestamp":1698908525000},"update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Convergence Rates of Proximal Alternating Direction Method of Multipliers"],"prefix":"10.1007","volume":"97","author":[{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0001-9283-9791","authenticated-orcid":false,"given":"Qinian","family":"Jin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,11,2]]},"reference":[{"key":"2383_CR1","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. Ser. B 116, 5\u201316 (2009)","journal-title":"Math. Program. Ser. B"},{"issue":"1","key":"2383_CR2","first-page":"17","volume":"5","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Soueycatt, M.: Augmented Lagrangian and proximal alternating direction methods of multipliers in Hilbert spaces. Applications to games, PDE\u2019s and control. Pac. J. Optim. 5(1), 17\u201337 (2009)","journal-title":"Pac. J. Optim."},{"key":"2383_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)"},{"key":"2383_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume-title":"First-Order Methods in Optimization, MOS-SIAM Series on Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-Order Methods in Optimization, MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2017)"},{"issue":"1","key":"2383_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/2200000016","volume":"3","author":"S Boyd","year":"2011","unstructured":"Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1\u2013122 (2011)","journal-title":"Found. Trends Mach. Learn."},{"issue":"3","key":"2383_CR6","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1007\/s10957-017-1112-5","volume":"173","author":"K Bredies","year":"2017","unstructured":"Bredies, K., Sun, H.: A proximal point analysis of the preconditioned alternating direction method of multipliers. J. Optim. Theory Appl. 173(3), 878\u2013907 (2017)","journal-title":"J. Optim. Theory Appl."},{"issue":"5","key":"2383_CR7","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1088\/0266-5611\/20\/5\/005","volume":"20","author":"M Burger","year":"2004","unstructured":"Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Prob. 20(5), 1411\u20131421 (2004)","journal-title":"Inverse Prob."},{"key":"2383_CR8","doi-asserted-by":"crossref","unstructured":"Davis, D., Yin, W.: Convergence rate analysis of several splitting schemes. In: Splitting Methods in Communication, Imaging, Science, and Engineering. Science Computing, pp. 115\u2013163. Springer, Cham (2016)","DOI":"10.1007\/978-3-319-41589-5_4"},{"key":"2383_CR9","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1007\/s10915-015-0048-x","volume":"66","author":"W Deng","year":"2016","unstructured":"Deng, W., Yin, W.: On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput. 66, 889\u2013916 (2016)","journal-title":"J. Sci. Comput."},{"issue":"1\u20133","key":"2383_CR10","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., Bertsekas, D.P.: On the Douglas\u2013Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1\u20133), 293\u2013318 (1992)","journal-title":"Math. Program."},{"key":"2383_CR11","volume-title":"Convex Analysis and Variational Problems","author":"I Ekeland","year":"1976","unstructured":"Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland, Amsterdam (1976)"},{"key":"2383_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-1740-8","volume-title":"Regularization of Inverse Problems","author":"HW Engl","year":"1996","unstructured":"Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)"},{"issue":"2","key":"2383_CR13","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1216\/JIE-2010-22-2-217","volume":"22","author":"K Frick","year":"2010","unstructured":"Frick, K., Scherzer, O.: Regularization of ill-posed linear equations by the non-stationary augmented Lagrangian method. J. Integral Equ. Appl. 22(2), 217\u2013257 (2010)","journal-title":"J. Integral Equ. Appl."},{"key":"2383_CR14","first-page":"299","volume":"15","author":"D Gabay","year":"1983","unstructured":"Gabay, D.: Applications of the method of multipliers to variational inequalities. Stud. Math. Appl. 15, 299\u2013331 (1983)","journal-title":"Stud. Math. Appl."},{"key":"2383_CR15","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."},{"issue":"R2","key":"2383_CR16","first-page":"41","volume":"9","author":"R Glowinski","year":"1975","unstructured":"Glowinski, R., Marrocco, 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\u00e9mes de Dirichlet non lin\u00e9aires. ESAIM Math. Model. Numer. Anal. 9(R2), 41\u201376 (1975)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"1","key":"2383_CR17","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1007\/s10915-009-9331-z","volume":"45","author":"T Goldstein","year":"2010","unstructured":"Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: segmentation and surface reconstruction. J. Sci. Comput. 45(1), 272\u2013293 (2010)","journal-title":"J. Sci. Comput."},{"key":"2383_CR18","volume-title":"The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics","author":"CW Groetsch","year":"1984","unstructured":"Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics, vol. 105. Pitman (Advanced Publishing Program), Boston, MA (1984)"},{"key":"2383_CR19","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1137\/110836936","volume":"50","author":"B He","year":"2012","unstructured":"He, B., Yuan, X.: On the $$O(1\/n)$$ convergence rate of Douglas\u2013Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700\u2013709 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"2383_CR20","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s00211-014-0673-6","volume":"130","author":"B He","year":"2015","unstructured":"He, B., Yuan, X.: On non-ergodic convergence rate of Douglas\u2013Rachford alternating direction method of multipliers. Numer. Math. 130, 567\u2013577 (2015)","journal-title":"Numer. Math."},{"issue":"4","key":"2383_CR21","doi-asserted-by":"publisher","first-page":"2114","DOI":"10.1137\/15M1029308","volume":"54","author":"Y Jiao","year":"2016","unstructured":"Jiao, Y., Jin, Q., Lu, X., Wang, W.: Alternating direction method of multipliers for linear inverse problems. SIAM J. Numer. Anal. 54(4), 2114\u20132137 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"2383_CR22","doi-asserted-by":"publisher","first-page":"025004","DOI":"10.1088\/1361-6420\/33\/2\/025004","volume":"33","author":"Y Jiao","year":"2017","unstructured":"Jiao, Y., Jin, Q., Lu, X., Wang, W.: Preconditioned alternating direction method of multipliers for solving inverse problems with constraints. Inverse Probl. 33, 025004 (2017)","journal-title":"Inverse Probl."},{"issue":"4","key":"2383_CR23","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1007\/s00211-022-01300-4","volume":"151","author":"Q Jin","year":"2022","unstructured":"Jin, Q.: Convergence rates of a dual gradient method for constrained linear ill-posed problems. Numer. Math. 151(4), 841\u2013871 (2022)","journal-title":"Numer. Math."},{"issue":"1","key":"2383_CR24","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1137\/20M1331901","volume":"32","author":"JH Lee","year":"2022","unstructured":"Lee, J.H., Pham, T.S.: Openness, H\u00f6lder metric regularity, and H\u00f6lder continuity properties of semialgebraic set-valued maps. SIAM J. Optim. 32(1), 56\u201374 (2022)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"2383_CR25","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s40305-015-0092-0","volume":"3","author":"T Lin","year":"2015","unstructured":"Lin, T., Ma, S., Zhang, S.: On the sublinear convergence rate of multi-block ADMM. J. Oper. Res. Soc. China 3(3), 251\u2013274 (2015)","journal-title":"J. Oper. Res. Soc. China"},{"issue":"6","key":"2383_CR26","doi-asserted-by":"publisher","first-page":"964","DOI":"10.1137\/0716071","volume":"16","author":"PL Lions","year":"1979","unstructured":"Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964\u2013979 (1979)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"2383_CR27","doi-asserted-by":"publisher","first-page":"2095","DOI":"10.1137\/17M1144623","volume":"56","author":"Y Liu","year":"2018","unstructured":"Liu, Y., Yuan, X., Zeng, S., Zhang, J.: Partial error bound conditions and the linear convergence rate of the alternating direction method of multipliers. SIAM J. Numer. Anal. 56(4), 2095\u20132123 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"2383_CR28","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719284","volume-title":"The Mathematics of Computerized Tomography","author":"F Natterer","year":"2001","unstructured":"Natterer, F.: The Mathematics of Computerized Tomography. SIAM, Philadelphia (2001)"},{"issue":"3","key":"2383_CR29","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1088\/0266-5611\/22\/3\/004","volume":"22","author":"E Resmerita","year":"2006","unstructured":"Resmerita, E., Scherzer, O.: Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Prob. 22(3), 801\u2013814 (2006)","journal-title":"Inverse Prob."},{"key":"2383_CR30","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1007\/BFb0120929","volume":"14","author":"SM Robinson","year":"1981","unstructured":"Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Study 14, 206\u2013214 (1981)","journal-title":"Math. Program. Study"},{"issue":"1","key":"2383_CR31","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s10957-019-01535-6","volume":"183","author":"H Sun","year":"2019","unstructured":"Sun, H.: Analysis of fully preconditioned alternating direction method of multipliers with relaxation in Hilbert spaces. J. Optim. Theory Appl. 183(1), 199\u2013229 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2383_CR32","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1137\/140974237","volume":"54","author":"WH Yang","year":"2016","unstructured":"Yang, W.H., Han, D.R.: Linear convergence of alternating direction method of multipliers for a class of convex optimization problems. SIAM J. Numer. Anal. 54(2), 625\u2013640 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"2383_CR33","doi-asserted-by":"publisher","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-dual algorithm framework based on Bregman iteration. J. Sci. Comput. 46, 20\u201346 (2011)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"2383_CR34","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1137\/120889502","volume":"24","author":"XY Zheng","year":"2014","unstructured":"Zheng, X.Y., Ng, K.F.: Metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization. SIAM J. Optim. 24(1), 154\u2013174 (2014)","journal-title":"SIAM J. Optim."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02383-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/article\/10.1007\/s10915-023-02383-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02383-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T20:08:34Z","timestamp":1700597314000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/10.1007\/s10915-023-02383-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,2]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["2383"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/s10915-023-02383-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,2]]},"assertion":[{"value":"13 March 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 September 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 November 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"66"}}