{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T18:58:47Z","timestamp":1776884327328,"version":"3.51.2"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T00:00:00Z","timestamp":1694649600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T00:00:00Z","timestamp":1694649600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100018542","name":"Natural Science Foundation of Sichuan Province","doi-asserted-by":"crossref","award":["2022NSFSC1830"],"award-info":[{"award-number":["2022NSFSC1830"]}],"id":[{"id":"10.13039\/501100018542","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Southwest Minzu University Research Startup Funds","award":["RQD2022035"],"award-info":[{"award-number":["RQD2022035"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,5]]},"DOI":"10.1007\/s11075-023-01645-3","type":"journal-article","created":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T13:51:47Z","timestamp":1694699507000},"page":"237-266","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Proximal variable smoothing method for three-composite nonconvex nonsmooth minimization with a linear operator"],"prefix":"10.1007","volume":"96","author":[{"given":"Yuncheng","family":"Liu","sequence":"first","affiliation":[]},{"given":"Fuquan","family":"Xia","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,9,14]]},"reference":[{"issue":"1","key":"1645_CR1","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. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(1), 91\u2013108 (2005)","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"1645_CR2","unstructured":"Ko, S., Won, J.-H.: Optimal minimization of the sum of three convex functions with a linear operator. In: The 22nd International Conference on Artificial Intelligence and Statistics, pp. 1185\u20131194 (2019)"},{"key":"1645_CR3","doi-asserted-by":"crossref","unstructured":"Zass, R., Shashua, A.: Nonnegative sparse pca. Advances in neural information processing systems 19 (2006)","DOI":"10.7551\/mitpress\/7503.003.0200"},{"issue":"115","key":"1645_CR4","first-page":"1","volume":"22","author":"MR Metel","year":"2021","unstructured":"Metel, M.R., Takeda, A.: Stochastic proximal methods for non-smooth non-convex constrained sparse optimization. J. Mach. Learn. Res. 22(115), 1\u201336 (2021)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"1645_CR5","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s10957-019-01477-z","volume":"181","author":"Y Liu","year":"2019","unstructured":"Liu, Y., Yin, W.: An envelope for davis-yin splitting and strict saddle-point avoidance. J. Optim. Theory Appl. 181(2), 567\u2013587 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"1645_CR6","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1007\/s11228-017-0421-z","volume":"25","author":"D Davis","year":"2017","unstructured":"Davis, D., Yin, W.: A three-operator splitting scheme and its optimization applications. Set-valued and variational analysis 25(4), 829\u2013858 (2017)","journal-title":"Set-valued and variational analysis"},{"issue":"4","key":"1645_CR7","doi-asserted-by":"publisher","first-page":"2809","DOI":"10.1137\/20M1326775","volume":"43","author":"F Bian","year":"2021","unstructured":"Bian, F., Zhang, X.: A three-operator splitting algorithm for nonconvex sparsity regularization. SIAM J. Sci. Comput. 43(4), 2809\u20132839 (2021)","journal-title":"SIAM J. Sci. Comput."},{"key":"1645_CR8","unstructured":"Yurtsever, A., Mangalick, V., Sra, S.: Three operator splitting with a nonconvex loss function. In: Proceedings of the 38th International Conference on Machine Learning, pp. 12267\u201312277 (2021)"},{"key":"1645_CR9","unstructured":"Zhao, R., Cevher, V.: Stochastic three-composite convex minimization with a linear operator. In: The 20nd International Conference on Artificial Intelligence and Statistics, pp. 765\u2013774 (2018)"},{"key":"1645_CR10","unstructured":"Zhao, R., Haskell, W.B., Tan, V.Y.: An optimal algorithm for stochastic three-composite optimization. In: The 22nd International Conference on Artificial Intelligence and Statistics, pp. 428\u2013437 (2019)"},{"issue":"1","key":"1645_CR11","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1007\/s11750-014-0326-z","volume":"23","author":"RI Bot","year":"2015","unstructured":"Bot, R.I., Hendrich, C.: A variable smoothing algorithm for solving convex optimization problems. TOP 23(1), 124\u2013150 (2015)","journal-title":"TOP"},{"key":"1645_CR12","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1007\/s10957-020-01800-z","volume":"188","author":"Variable smoothing for weakly convex composite functions","year":"2021","unstructured":"Variable smoothing for weakly convex composite functions: B$$\\ddot{\\rm o }$$hm, A., Wright, S.J. J. Optim. Theory Appl. 188, 628\u2013649 (2021)","journal-title":"J. Optim. Theory Appl."},{"issue":"33","key":"1645_CR13","first-page":"1","volume":"85","author":"RI Bot","year":"2020","unstructured":"Bot, R.I., B\u00f6hm, A.: Variable smoothing for convex optimization problems using stochastic gradients. J. Sci. Comput. 85(33), 1\u201329 (2020)","journal-title":"J. Sci. Comput."},{"issue":"6","key":"1645_CR14","doi-asserted-by":"publisher","first-page":"2147","DOI":"10.1007\/s11590-021-01723-2","volume":"15","author":"Y Liu","year":"2021","unstructured":"Liu, Y., Xia, F.: Variable smoothing incremental aggregated gradient method for nonsmooth nonconvex regularized optimization. Optimization Letters 15(6), 2147\u20132164 (2021)","journal-title":"Optimization Letters"},{"key":"1645_CR15","doi-asserted-by":"crossref","unstructured":"Bertsekas, D.P.: Incremental gradient, subgradient, and proximal methods for convex optimization: A survey. Optimization for Machine Learning, 1\u201338 (2011)","DOI":"10.7551\/mitpress\/8996.003.0006"},{"issue":"4","key":"1645_CR16","doi-asserted-by":"publisher","first-page":"2542","DOI":"10.1137\/17M1147846","volume":"29","author":"M G\u00fcrb\u00fczbalaban","year":"2018","unstructured":"G\u00fcrb\u00fczbalaban, M., Ozdaglar, A.E., Parrilo, P.A.: Convergence rate of incremental gradient and incremental newton methods. SIAM J. Optim. 29(4), 2542\u20132565 (2018)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1645_CR17","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1137\/S1052623499362111","volume":"12","author":"A Nedi\u0107","year":"2001","unstructured":"Nedi\u0107, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12(1), 109\u2013138 (2001)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1645_CR18","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":"2","key":"1645_CR19","doi-asserted-by":"publisher","first-page":"1035","DOI":"10.1137\/15M1049695","volume":"27","author":"M G\u00fcrb\u00fczbalaban","year":"2017","unstructured":"G\u00fcrb\u00fczbalaban, M., Ozdaglar, A., Parrilo, P.A.: On the convergence rate of incremental aggregated gradient algorithms. SIAM J. Optim. 27(2), 1035\u20131048 (2017)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1645_CR20","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1007\/s10589-020-00183-1","volume":"76","author":"H-T Wai","year":"2020","unstructured":"Wai, H.-T., Shi, W., Uribe, C.A., Nedi\u0107, A., Scaglione, A.: Accelerating incremental gradient optimization with curvature information. Comput. Optim. Appl. 76(2), 347\u2013380 (2020)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"1645_CR21","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1287\/moor.2019.1047","volume":"46","author":"H Zhang","year":"2020","unstructured":"Zhang, H., Dai, Y.H., Guo, L.: Proximal-like incremental aggregated gradient method with linear convergence under bregman distance growth conditions. Math. Oper. Res. 46(1), 61\u201381 (2020)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"1645_CR22","doi-asserted-by":"publisher","first-page":"1282","DOI":"10.1137\/16M1094415","volume":"28","author":"ND Vanli","year":"2018","unstructured":"Vanli, N.D., G\u00fcrb\u00fczbalaban, M., Ozdaglar, A.: Global convergence rate of proximal incremental aggregated gradient methods. SIAM J. Optim. 28(2), 1282\u20131300 (2018)","journal-title":"SIAM J. Optim."},{"issue":"9","key":"1645_CR23","doi-asserted-by":"publisher","first-page":"3445","DOI":"10.1080\/00036811.2020.1849634","volume":"101","author":"Y Liu","year":"2022","unstructured":"Liu, Y., Xia, F.: Linear convergence of proximal incremental aggregated gradient method for nonconvex nonsmooth minimization problems. Appl. Anal. 101(9), 3445\u20133464 (2022)","journal-title":"Appl. Anal."},{"key":"1645_CR24","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1007\/s10957-019-01538-3","volume":"183","author":"W Peng","year":"2019","unstructured":"Peng, W., Zhang, H., Zhang, X.: Nonconvex proximal incremental aggregated gradient method with linear convergence. J. Optim. Theory Appl. 183, 230\u2013245 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1645_CR25","doi-asserted-by":"publisher","first-page":"1420","DOI":"10.1137\/16M1101702","volume":"28","author":"A Mokhtari","year":"2018","unstructured":"Mokhtari, A., G\u00fcrb\u00fczbalaban, M., Ribeiro, A.: Surpassing gradient descent provably: A cyclic incremental method with linear convergence rate. SIAM J. Optim. 28(2), 1420\u20131447 (2018)","journal-title":"SIAM J. Optim."},{"key":"1645_CR26","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (2009)"},{"key":"1645_CR27","doi-asserted-by":"crossref","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I, Volume 330 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, Heidelberg (2006)","DOI":"10.1007\/3-540-31247-1"},{"key":"1645_CR28","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume-title":"First-order Methods in Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-order Methods in Optimization. SIAM, Philadelphia (2017)"},{"key":"1645_CR29","doi-asserted-by":"crossref","unstructured":"Reddi, S.J., Hefny, A., Sra, S., Poczos, B., Smola, A.: Stochastic variance reduction for nonconvex optimization. In: Proceedings of the 32th International Conference on Machine Learning (2016)","DOI":"10.1109\/ALLERTON.2016.7852377"},{"issue":"4","key":"1645_CR30","doi-asserted-by":"publisher","first-page":"2341","DOI":"10.1137\/120880811","volume":"23","author":"S Ghadimi","year":"2013","unstructured":"Ghadimi, S., Lan, G.: Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM J. Optim. 23(4), 2341\u20132368 (2013)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"1645_CR31","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s10107-014-0846-1","volume":"155","author":"S Ghadimi","year":"2016","unstructured":"Ghadimi, S., Lan, G., Zhang, H.: Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization. Math. Program. 155(1\u20132), 267\u2013305 (2016)","journal-title":"Math. Program."},{"key":"1645_CR32","unstructured":"Li, Z., Jian, L.: A simple proximal stochastic gradient method for nonsmooth nonconvex optimization. In: The 32nd Conference on Neural Information Processing Systems, pp. 5564\u20135574 (2018)"},{"issue":"5","key":"1645_CR33","first-page":"1191","volume":"21","author":"K Tu","year":"2020","unstructured":"Tu, K., Zhang, H.B.: Gao H: Stochastic proximal difference-of-convex algorithm with spider for a class of nonconvex nonsmooth regularized problems. Journal of Nonlinear Convex and Analysis 21(5), 1191\u20131208 (2020)","journal-title":"Journal of Nonlinear Convex and Analysis"},{"issue":"3","key":"1645_CR34","doi-asserted-by":"publisher","first-page":"2158","DOI":"10.1137\/17M1141849","volume":"28","author":"HH Bauschke","year":"2018","unstructured":"Bauschke, H.H., Bui, M.N., Wang, X.: Projecting onto the intersection of a cone and a sphere. SIAM J. Optim. 28(3), 2158\u20132188 (2018)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1645_CR35","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"CH Zhang","year":"2010","unstructured":"Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"issue":"456","key":"1645_CR36","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348\u20131360 (2001)","journal-title":"J. Am. Stat. Assoc."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01645-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01645-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01645-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T10:10:52Z","timestamp":1712311852000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01645-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,14]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["1645"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01645-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,14]]},"assertion":[{"value":"20 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval and consent to participate"}},{"value":"Not applicable","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"Not applicable","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and animal ethics"}},{"value":"Not applicable","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}