{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T16:11:44Z","timestamp":1789143104185,"version":"build-2803163510"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T00:00:00Z","timestamp":1787961600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T00:00:00Z","timestamp":1787961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801279"],"award-info":[{"award-number":["11801279"]}],"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":["BK20180782"],"award-info":[{"award-number":["BK20180782"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["12131004"],"award-info":[{"award-number":["12131004"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1007\/s10915-026-03434-1","type":"journal-article","created":{"date-parts":[[2026,8,29]],"date-time":"2026-08-29T08:12:33Z","timestamp":1787991153000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Variance Reduced Douglas-Rachford Splitting Method for Large-Scale Optimization Problems"],"prefix":"10.1007","volume":"109","author":[{"given":"Zehui","family":"Jia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Denghui","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiyu","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0427-8224","authenticated-orcid":false,"given":"Deren","family":"Han","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,29]]},"reference":[{"issue":"1","key":"3434_CR1","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods. Math. Program. 137(1), 91\u2013129 (2013)","journal-title":"Math. Program."},{"issue":"1","key":"3434_CR2","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1137\/040615961","volume":"18","author":"D Blatt","year":"2007","unstructured":"Blatt, D., Hero, A.O., Gauchman, H.: A convergent incremental gradient method with a constant step size. SIAM J. Optim. 18(1), 29\u201351 (2007)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"3434_CR3","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1), 459\u2013494 (2014)","journal-title":"Math. Program."},{"issue":"3","key":"3434_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1961189.1961199","volume":"2","author":"C-C Chang","year":"2011","unstructured":"Chang, C.-C., Lin, C.-J.: Libsvm: A library for support vector machines. ACM transactions on intelligent systems and technology (TIST) 2(3), 1\u201327 (2011)","journal-title":"ACM transactions on intelligent systems and technology (TIST)"},{"issue":"2","key":"3434_CR5","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1137\/140971233","volume":"25","author":"PL Combettes","year":"2015","unstructured":"Combettes, P.L., Pesquet, J.-C.: Stochastic quasi-fej\u00e9r block-coordinate fixed point iterations with random sweeping. SIAM J. Optim. 25(2), 1221\u20131248 (2015)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3434_CR6","doi-asserted-by":"publisher","first-page":"1693","DOI":"10.1137\/18M1231559","volume":"30","author":"Y Cui","year":"2020","unstructured":"Cui, Y., He, Z., Pang, J.-S.: Multicomposite nonconvex optimization for training deep neural networks. SIAM J. Optim. 30(2), 1693\u20131723 (2020)","journal-title":"SIAM J. Optim."},{"key":"3434_CR7","unstructured":"Defazio, A.: A simple practical accelerated method for finite sums, Advances in neural information processing systems 29, (2016)"},{"key":"3434_CR8","unstructured":"Defazio, A., Bach, F., Lacoste-Julien, S.: Saga: A fast incremental gradient method with support for non-strongly convex composite objectives. In: Advances in neural information processing systems, 27, (2014)"},{"issue":"4","key":"3434_CR9","doi-asserted-by":"publisher","first-page":"1691","DOI":"10.1137\/23M1622490","volume":"63","author":"Y Dong","year":"2025","unstructured":"Dong, Y., Bredies, K., Sun, H.: A stochastic preconditioned douglas-rachford splitting method for saddle-point problems. SIAM J. Numer. Anal. 63(4), 1691\u20131728 (2025)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"3434_CR10","doi-asserted-by":"publisher","first-page":"1932","DOI":"10.1137\/20M1387213","volume":"14","author":"D Driggs","year":"2021","unstructured":"Driggs, D., Tang, J., Liang, J., Davies, M., Schonlieb, C.-B.: A stochastic proximal alternating minimization for nonsmooth and nonconvex optimization. SIAM J. Imag. Sci. 14(4), 1932\u20131970 (2021)","journal-title":"SIAM J. Imag. Sci."},{"issue":"1","key":"3434_CR11","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1137\/23M1567503","volume":"34","author":"D Han","year":"2024","unstructured":"Han, D., Su, Y., Xie, J.: Randomized douglas-rachford methods for linear systems: Improved accuracy and efficiency. SIAM J. Optim. 34(1), 1045\u20131070 (2024)","journal-title":"SIAM J. Optim."},{"key":"3434_CR12","doi-asserted-by":"crossref","unstructured":"Reddi, S.J., Sra, S., Poczos, B., Smola, A.J.: Proximal stochastic methods for nonsmooth nonconvex finite-sum optimization. Adv. Neural. Inf. Process. Syst. 29, (2016)","DOI":"10.1109\/CDC.2016.7798553"},{"key":"3434_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.113384","volume":"390","author":"Z Jia","year":"2021","unstructured":"Jia, Z., Huang, J., Wu, Z.: An incremental aggregated proximal admm for linearly constrained nonconvex optimization with application to sparse logistic regression problems. J. Comput. Appl. Math. 390, 113384 (2021)","journal-title":"J. Comput. Appl. Math."},{"issue":"348","key":"3434_CR14","doi-asserted-by":"publisher","first-page":"1677","DOI":"10.1090\/mcom\/3867","volume":"93","author":"Z Jia","year":"2024","unstructured":"Jia, Z., Zhang, W., Cai, X., Han, D.: Stochastic alternating structure-adapted proximal gradient descent method with variance reduction for nonconvex nonsmooth optimization. Math. Comput. 93(348), 1677\u20131714 (2024)","journal-title":"Math. Comput."},{"key":"3434_CR15","unstructured":"Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction. In: Advances in neural information processing systems, 26, (2013)"},{"issue":"2","key":"3434_CR16","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/s10589-017-9915-8","volume":"68","author":"G Li","year":"2017","unstructured":"Li, G., Liu, T., Pong, T.K.: Peaceman-rachford splitting for a class of nonconvex optimization problems. Comput. Optim. Appl. 68(2), 407\u2013436 (2017)","journal-title":"Comput. Optim. Appl."},{"issue":"4","key":"3434_CR17","doi-asserted-by":"publisher","first-page":"2434","DOI":"10.1137\/140998135","volume":"25","author":"G Li","year":"2015","unstructured":"Li, G., Pong, T.K.: Global convergence of splitting methods for nonconvex composite optimization. SIAM J. Optim. 25(4), 2434\u20132460 (2015)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"3434_CR18","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s10107-015-0963-5","volume":"159","author":"G Li","year":"2016","unstructured":"Li, G., Pong, T.K.: Douglas-rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Math. Program. 159(1), 371\u2013401 (2016)","journal-title":"Math. Program."},{"issue":"6","key":"3434_CR19","doi-asserted-by":"publisher","first-page":"964","DOI":"10.1137\/0716071","volume":"16","author":"P-L 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":"5","key":"3434_CR20","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1007\/s10994-019-05785-3","volume":"108","author":"J Liu","year":"2019","unstructured":"Liu, J., Xu, L., Shen, S., Ling, Q.: An accelerated variance reducing stochastic method with douglas-rachford splitting. Mach. Learn. 108(5), 859\u2013878 (2019)","journal-title":"Mach. Learn."},{"key":"3434_CR21","unstructured":"Nesterov, Y.: Introductory lectures on convex optimization: A basic course, vol. 87. Springer Science & Business Media (2013)"},{"key":"3434_CR22","doi-asserted-by":"crossref","unstructured":"Nesterov, Y., et al.: Lectures on convex optimization, vol 137. Springer, (2018)","DOI":"10.1007\/978-3-319-91578-4_2"},{"key":"3434_CR23","doi-asserted-by":"crossref","unstructured":"Patrinos, P., Stella, L., Bemporad, A.: Douglas-rachford splitting: Complexity estimates and accelerated variants. In 53rd IEEE Conference on Decision and Control, IEEE, pp 4234\u20134239. (2014)","DOI":"10.1109\/CDC.2014.7040049"},{"issue":"110","key":"3434_CR24","first-page":"1","volume":"21","author":"NH Pham","year":"2020","unstructured":"Pham, N.H., Nguyen, L.M., Phan, D.T., Tran-Dinh, Q.: Proxsarah: An efficient algorithmic framework for stochastic composite nonconvex optimization. J. Mach. Learn. Res. 21(110), 1\u201348 (2020)","journal-title":"J. Mach. Learn. Res."},{"key":"3434_CR25","doi-asserted-by":"crossref","unstructured":"Robbins, H., Monro, S.: A stochastic approximation method. The annals of mathematical statistics, pp 400\u2013407 (1951)","DOI":"10.1214\/aoms\/1177729586"},{"issue":"5","key":"3434_CR26","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14(5), 877\u2013898 (1976)","journal-title":"SIAM J. Control. Optim."},{"key":"3434_CR27","unstructured":"Rockafellar, R.T., Wets, R. J.-B.: Variational Analysis, vol 317. Springer Science & Business Media (2009)"},{"key":"3434_CR28","unstructured":"Rudin, W.: Real and complex analysis, McGraw-Hill Inc (1987)"},{"issue":"7","key":"3434_CR29","first-page":"621","volume":"20","author":"J Shermen","year":"1949","unstructured":"Shermen, J., Morrison, W.: Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix. Annual Mathmatical Statistics 20(7), 621\u2013625 (1949)","journal-title":"Annual Mathmatical Statistics"},{"issue":"1","key":"3434_CR30","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1137\/18M1163993","volume":"30","author":"A Themelis","year":"2020","unstructured":"Themelis, A., Patrinos, P.: Douglas-rachford splitting and admm for nonconvex optimization: Tight convergence results. SIAM J. Optim. 30(1), 149\u2013181 (2020)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3434_CR31","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s10589-022-00366-y","volume":"82","author":"A Themelis","year":"2022","unstructured":"Themelis, A., Stella, L., Patrinos, P.: Douglas-rachford splitting and admm for nonconvex optimization: accelerated and newton-type linesearch algorithms. Comput. Optim. Appl. 82(2), 395\u2013440 (2022)","journal-title":"Comput. Optim. Appl."},{"issue":"4","key":"3434_CR32","doi-asserted-by":"publisher","first-page":"2057","DOI":"10.1137\/140961791","volume":"24","author":"L Xiao","year":"2014","unstructured":"Xiao, L., Zhang, T.: A proximal stochastic gradient method with progressive variance reduction. SIAM J. Optim. 24(4), 2057\u20132075 (2014)","journal-title":"SIAM J. Optim."},{"issue":"7","key":"3434_CR33","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1109\/TNNLS.2012.2197412","volume":"23","author":"Z Xu","year":"2012","unstructured":"Xu, Z., Chang, X., Xu, F., Zhang, H.: $$ l_\\frac{1}{2}$$ regularization: A thresholding representation theory and a fast solver. IEEE Transactions on neural networks and learning systems 23(7), 1013\u20131027 (2012)","journal-title":"IEEE Transactions on neural networks and learning systems"},{"key":"3434_CR34","unstructured":"Yuan, H., Zaheer, M., Reddi, S.: Federated composite optimization. In International Conference on Machine Learning, PMLR, pp 12253\u201312266 (2021)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03434-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03434-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03434-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T15:45:08Z","timestamp":1789141508000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03434-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,29]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["3434"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03434-1","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,29]]},"assertion":[{"value":"4 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 August 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"30"}}