{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T00:42:06Z","timestamp":1762389726309,"version":"3.37.3"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T00:00:00Z","timestamp":1601683200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T00:00:00Z","timestamp":1601683200000},"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":["11871269"],"award-info":[{"award-number":["11871269"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20181259"],"award-info":[{"award-number":["BK20181259"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"crossref","award":["17BTQ063"],"award-info":[{"award-number":["17BTQ063"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100018562","name":"Social Science Foundation of Jiangsu Province","doi-asserted-by":"crossref","award":["18GLA002","17ZTB011"],"award-info":[{"award-number":["18GLA002","17ZTB011"]}],"id":[{"id":"10.13039\/501100018562","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012456","name":"National Social Science Foundation of China","doi-asserted-by":"crossref","award":["19BGL205","19AZD018"],"award-info":[{"award-number":["19BGL205","19AZD018"]}],"id":[{"id":"10.13039\/501100012456","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2021,1]]},"DOI":"10.1007\/s10589-020-00229-4","type":"journal-article","created":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T10:02:42Z","timestamp":1601719362000},"page":"87-124","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Convergence study on strictly contractive Peaceman\u2013Rachford splitting method for nonseparable convex minimization models with quadratic coupling terms"],"prefix":"10.1007","volume":"78","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3792-7059","authenticated-orcid":false,"given":"Peixuan","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Shen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Suhong","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zehua","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Caihua","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,10,3]]},"reference":[{"issue":"1","key":"229_CR1","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s10589-017-9971-0","volume":"70","author":"J Bai","year":"2018","unstructured":"Bai, J., Li, J., Xu, F., Zhang, H.: Generalized symmetric ADMM for separable convex optimization. Comput. Optim. Appl. 70(1), 129\u2013170 (2018)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"229_CR2","first-page":"1","volume":"3","author":"S Boyd","year":"2011","unstructured":"Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J., et al.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends\u00ae Mach. Learn. 3(1), 1\u2013122 (2011)","journal-title":"Found. Trends\u00ae Mach. Learn."},{"key":"229_CR3","volume-title":"Finite Element Approximation and Iterative Solution of a Class of Mildly Non-linear Elliptic Equations","author":"TFC Chan","year":"1978","unstructured":"Chan, T.F.C., Glowinski, R.: Finite Element Approximation and Iterative Solution of a Class of Mildly Non-linear Elliptic Equations. Computer Science Department, Stanford University, Stanford (1978)"},{"issue":"1\u20132","key":"229_CR4","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10107-014-0826-5","volume":"155","author":"C Chen","year":"2016","unstructured":"Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math. Program. 155(1\u20132), 57\u201379 (2016)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"229_CR5","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10107-017-1205-9","volume":"173","author":"C Chen","year":"2019","unstructured":"Chen, C., Li, M., Liu, X., Ye, Y.: Extended ADMM and BCD for nonseparable convex minimization models with quadratic coupling terms: convergence analysis and insights. Math. Program. 173(1\u20132), 37\u201377 (2019)","journal-title":"Math. Program."},{"issue":"4","key":"229_CR6","doi-asserted-by":"publisher","first-page":"2120","DOI":"10.1137\/140980910","volume":"25","author":"C Chen","year":"2015","unstructured":"Chen, C., Ma, S., Yang, J.: A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J. Optim. 25(4), 2120\u20132142 (2015)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"229_CR7","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1137\/S003614450037906X","volume":"43","author":"SS Chen","year":"2001","unstructured":"Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129\u2013159 (2001)","journal-title":"SIAM Rev."},{"issue":"4","key":"229_CR8","doi-asserted-by":"publisher","first-page":"986","DOI":"10.1137\/040613305","volume":"16","author":"YH Dai","year":"2006","unstructured":"Dai, Y.H.: Fast algorithms for projection on an ellipsoid. SIAM J. Optim. 16(4), 986\u20131006 (2006)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"229_CR9","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1007\/s10915-016-0318-2","volume":"71","author":"W Deng","year":"2017","unstructured":"Deng, W., Lai, M.J., Peng, Z., Yin, W.: Parallel multi-block ADMM with $$o(1\/k)$$ convergence. J. Sci. Comput. 71(2), 712\u2013736 (2017)","journal-title":"J. Sci. Comput."},{"issue":"1\u20133","key":"229_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."},{"issue":"3","key":"229_CR11","first-page":"44","volume":"32","author":"J Eckstein","year":"2012","unstructured":"Eckstein, J., Yao, W.: Augmented Lagrangian and alternating direction methods for convex optimization: a tutorial and some illustrative computational results. RUTCOR Res. Rep. 32(3), 44 (2012)","journal-title":"RUTCOR Res. Rep."},{"issue":"1","key":"229_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(1), 17\u201340 (1976)","journal-title":"Comput. Math. Appl."},{"key":"229_CR13","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/978-94-017-9054-3_4","volume-title":"Modeling, Simulation and Optimization for Science and Technology","author":"R Glowinski","year":"2014","unstructured":"Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. In: Fitzgibbon, W., Kuznetsov, Y. A., Neittaanm\u00e4ki, P., Pironneau, O. (eds.) Modeling, Simulation and Optimization for Science and Technology, pp. 59\u201382. Springer, Dordrecht (2014)"},{"key":"229_CR14","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970838","volume-title":"Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics","author":"R Glowinski","year":"1989","unstructured":"Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9. SIAM, Philadelphia (1989)"},{"issue":"R2","key":"229_CR15","first-page":"41","volume":"9","author":"R Glowinski","year":"1975","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(R2), 41\u201376 (1975)","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis-Mod\u00e9lisation Math\u00e9matique et Analyse Num\u00e9rique"},{"key":"229_CR16","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1115\/1.3169136","volume":"52","author":"R Glowinski","year":"1985","unstructured":"Glowinski, R., Oden, J.T.: Numerical methods for nonlinear variational problems. J. Appl. Mech. 52, 739 (1985)","journal-title":"J. Appl. Mech."},{"issue":"1","key":"229_CR17","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s10957-013-0489-z","volume":"163","author":"G Gu","year":"2014","unstructured":"Gu, G., He, B., Yang, J.: Inexact alternating-direction-based contraction methods for separable linearly constrained convex optimization. J. Optim. Theory Appl. 163(1), 105\u2013129 (2014)","journal-title":"J. Optim. Theory Appl."},{"key":"229_CR18","unstructured":"Gu, Y., Jiang, B., Han, D.: A semi-proximal-based strictly contractive Peaceman\u2013Rachford splitting method. arXiv preprint arXiv:1506.02221 (2015)"},{"issue":"2","key":"229_CR19","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s10589-012-9510-y","volume":"54","author":"D Han","year":"2013","unstructured":"Han, D., Yuan, X., Zhang, W., Cai, X.: An ADM-based splitting method for separable convex programming. Comput. Optim. Appl. 54(2), 343\u2013369 (2013)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"229_CR20","doi-asserted-by":"publisher","first-page":"1011","DOI":"10.1137\/13090849X","volume":"24","author":"B He","year":"2014","unstructured":"He, B., Liu, H., Wang, Z., Yuan, X.: A strictly contractive Peaceman\u2013Rachford splitting method for convex programming. SIAM J. Optim. 24(3), 1011\u20131040 (2014)","journal-title":"SIAM J. Optim."},{"key":"229_CR21","unstructured":"He, B., Ma, F., Yuan, X.: On the step size of symmetric alternating directions method of multipliers (2015). Preprint at: http:\/\/www.optimization-online.org\/DB_FILE\/2015\/05\/4925.pdf. Accessed 30 Sep 2020"},{"issue":"3","key":"229_CR22","doi-asserted-by":"publisher","first-page":"1467","DOI":"10.1137\/15M1044448","volume":"9","author":"B He","year":"2016","unstructured":"He, B., Ma, F., Yuan, X.: Convergence study on the symmetric version of ADMM with larger step sizes. SIAM J. Imaging Sci. 9(3), 1467\u20131501 (2016)","journal-title":"SIAM J. Imaging Sci."},{"issue":"2","key":"229_CR23","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1137\/110822347","volume":"22","author":"B He","year":"2012","unstructured":"He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2), 313\u2013340 (2012)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"229_CR24","doi-asserted-by":"publisher","first-page":"662","DOI":"10.1287\/moor.2016.0822","volume":"42","author":"B He","year":"2017","unstructured":"He, B., Tao, M., Yuan, X.: Convergence rate analysis for the alternating direction method of multipliers with a substitution procedure for separable convex programming. Math. Oper. Res. 42(3), 662\u2013691 (2017)","journal-title":"Math. Oper. Res."},{"issue":"3","key":"229_CR25","doi-asserted-by":"publisher","first-page":"1204","DOI":"10.1007\/s10915-015-0060-1","volume":"66","author":"B He","year":"2016","unstructured":"He, B., Xu, H.K., Yuan, X.: On the proximal Jacobian decomposition of ALM for multiple-block separable convex minimization problems and its relationship to ADMM. J. Sci. Comput. 66(3), 1204\u20131217 (2016)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"229_CR26","doi-asserted-by":"crossref","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 the Douglas\u2013Rachford alternating direction method. SIAM J. Numer. Anal. 50(2), 700\u2013709 (2012)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"229_CR27","doi-asserted-by":"publisher","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(5), 303\u2013320 (1969)","journal-title":"J. Optim. Theory Appl."},{"issue":"1\u20132","key":"229_CR28","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s10107-016-1034-2","volume":"162","author":"M Hong","year":"2017","unstructured":"Hong, M., Luo, Z.Q.: On the linear convergence of the alternating direction method of multipliers. Math. Program. 162(1\u20132), 165\u2013199 (2017)","journal-title":"Math. Program."},{"key":"229_CR29","doi-asserted-by":"publisher","first-page":"320","DOI":"10.1016\/j.cam.2017.01.008","volume":"319","author":"Z Jia","year":"2017","unstructured":"Jia, Z., Cai, X., Han, D.: Comparison of several fast algorithms for projection onto an ellipsoid. J. Comput. Appl. Math. 319, 320\u2013337 (2017)","journal-title":"J. Comput. Appl. Math."},{"issue":"04","key":"229_CR30","doi-asserted-by":"publisher","first-page":"1550024","DOI":"10.1142\/S0217595915500244","volume":"32","author":"M Li","year":"2015","unstructured":"Li, M., Sun, D., Toh, K.C.: A convergent 3-block semi-proximal ADMM for convex minimization problems with one strongly convex block. Asia Pac. J. Oper. Res. 32(04), 1550024 (2015)","journal-title":"Asia Pac. J. Oper. Res."},{"issue":"6","key":"229_CR31","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."},{"key":"229_CR32","doi-asserted-by":"publisher","first-page":"76131","DOI":"10.1109\/ACCESS.2019.2914461","volume":"7","author":"Q Liu","year":"2019","unstructured":"Liu, Q., Shen, X., Gu, Y.: Linearized ADMM for nonconvex nonsmooth optimization with convergence analysis. IEEE Access 7, 76131\u201376144 (2019). https:\/\/doi.org\/10.1109\/ACCESS.2019.2914461","journal-title":"IEEE Access"},{"issue":"1","key":"229_CR33","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1137\/0103003","volume":"3","author":"DW Peaceman","year":"1955","unstructured":"Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28\u201341 (1955)","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"229_CR34","volume-title":"Optimization","author":"MJ Powell","year":"1969","unstructured":"Powell, M.J.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.), Optimization, pp. 283\u2013298. Academic Press, New York, NY (1969)"},{"issue":"06","key":"229_CR35","doi-asserted-by":"publisher","first-page":"1750030","DOI":"10.1142\/S0217595917500300","volume":"34","author":"Z Wu","year":"2017","unstructured":"Wu, Z., Li, M., Wang, D.Z., Han, D.: A symmetric alternating direction method of multipliers for separable nonconvex minimization problems. Asia Pac. J. Oper. Res. 34(06), 1750030 (2017)","journal-title":"Asia Pac. J. Oper. Res."},{"issue":"281","key":"229_CR36","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1090\/S0025-5718-2012-02598-1","volume":"82","author":"J Yang","year":"2013","unstructured":"Yang, J., Yuan, X.: Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization. Math. Comput. 82(281), 301\u2013329 (2013)","journal-title":"Math. Comput."},{"key":"229_CR37","first-page":"167","volume":"9","author":"X Yuan","year":"2013","unstructured":"Yuan, X., Yang, J.: Sparse and low-rank matrix decomposition via alternating direction methods. Pac. J. Optim. 9, 167\u2013180 (2013)","journal-title":"Pac. J. Optim."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00229-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-020-00229-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00229-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,8]],"date-time":"2023-10-08T23:54:45Z","timestamp":1696809285000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-020-00229-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,3]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["229"],"URL":"https:\/\/doi.org\/10.1007\/s10589-020-00229-4","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2020,10,3]]},"assertion":[{"value":"26 September 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 October 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}