{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T07:16:07Z","timestamp":1758093367763,"version":"3.44.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,8,30]],"date-time":"2025-08-30T00:00:00Z","timestamp":1756512000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,30]],"date-time":"2025-08-30T00:00:00Z","timestamp":1756512000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Shanghai Institute for Mathematics and Interdisciplinary Sciences","award":["Startup Fund"],"award-info":[{"award-number":["Startup Fund"]}]},{"DOI":"10.13039\/501100014761","name":"Natural Science Foundation of Qingdao Municipality","doi-asserted-by":"publisher","award":["23-2-1- 158-zyyd-jch"],"award-info":[{"award-number":["23-2-1- 158-zyyd-jch"]}],"id":[{"id":"10.13039\/501100014761","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12241105"],"award-info":[{"award-number":["12241105"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10957-025-02835-w","type":"journal-article","created":{"date-parts":[[2025,8,30]],"date-time":"2025-08-30T18:15:47Z","timestamp":1756577747000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Lyapunov Analysis of Accelerated PDHG Algorithms"],"prefix":"10.1007","volume":"207","author":[{"given":"Xueying","family":"Zeng","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8616-6180","authenticated-orcid":false,"given":"Bin","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,30]]},"reference":[{"key":"2835_CR1","unstructured":"Arrow, K.J., Hurwicz, L., Uzawa, H.: Studies in Linear and Non-Linear Programming. Vol. II of Stanford Mathematical Studies in The Social Sciences. Stanford University Press (1958)"},{"key":"2835_CR2","doi-asserted-by":"crossref","unstructured":"Beck, A.: First-Order Methods in Optimization. SIAM (2017)","DOI":"10.1137\/1.9781611974997"},{"key":"2835_CR3","doi-asserted-by":"crossref","unstructured":"Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)","DOI":"10.1017\/CBO9780511804441"},{"issue":"1","key":"2835_CR4","doi-asserted-by":"publisher","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-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40(1), 120\u2013145 (2011)","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2835_CR5","doi-asserted-by":"publisher","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"},{"issue":"1\u20132","key":"2835_CR6","doi-asserted-by":"publisher","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-dual algorithm. Math. Program. 159(1\u20132), 253\u2013287 (2016)","journal-title":"Math. Program."},{"issue":"1","key":"2835_CR7","doi-asserted-by":"publisher","first-page":"53","DOI":"10.3934\/ipi.2014.8.53","volume":"8","author":"F Chen","year":"2014","unstructured":"Chen, F., Shen, L., Xu, Y., Zeng, X.: The Moreau envelope approach for the L1\/TV image denoising model. Inverse Problems and Imaging 8(1), 53\u201377 (2014)","journal-title":"Inverse Problems and Imaging"},{"key":"2835_CR8","unstructured":"Chen, S., Shi, B., Yuan, Y.X.: Gradient norm minimization of Nesterov acceleration: $$ \\rm o\\it (1\/k^3) $$. arXiv preprint arXiv:2209.08862 (2022)"},{"key":"2835_CR9","unstructured":"Chen, S., Shi, B., Yuan, Y.X.: Revisiting the acceleration phenomenon via high-resolution differential equations. arXiv preprint arXiv:2212.05700 (2022)"},{"key":"2835_CR10","unstructured":"Chen, S., Shi, B., Yuan, Y.X.: On underdamped Nesterov\u2019s acceleration. arXiv preprint arXiv:2304.14642 (2023)"},{"issue":"432","key":"2835_CR11","doi-asserted-by":"publisher","first-page":"1200","DOI":"10.1080\/01621459.1995.10476626","volume":"90","author":"DL Donoho","year":"1995","unstructured":"Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200\u20131224 (1995)","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"2835_CR12","doi-asserted-by":"publisher","first-page":"2274","DOI":"10.1137\/130922793","volume":"25","author":"B He","year":"2015","unstructured":"He, B., Hou, L., Yuan, X.: On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming. SIAM J. Optim. 25(4), 2274\u20132312 (2015)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"2835_CR13","doi-asserted-by":"publisher","first-page":"1157","DOI":"10.1137\/21M1453463","volume":"15","author":"B He","year":"2022","unstructured":"He, B., Ma, F., Xu, S., Yuan, X.: A generalized primal-dual algorithm with improved convergence condition for saddle point problems. SIAM J. Imag. Sci. 15(3), 1157\u20131183 (2022)","journal-title":"SIAM J. Imag. Sci."},{"issue":"4","key":"2835_CR14","doi-asserted-by":"publisher","first-page":"2526","DOI":"10.1137\/140963467","volume":"7","author":"B He","year":"2014","unstructured":"He, B., You, Y., Yuan, X.: On the convergence of primal-dual hybrid gradient algorithm. SIAM J. Imag. Sci. 7(4), 2526\u20132537 (2014)","journal-title":"SIAM J. Imag. Sci."},{"issue":"1","key":"2835_CR15","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1137\/100814494","volume":"5","author":"B He","year":"2012","unstructured":"He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imag. Sci. 5(1), 119\u2013149 (2012)","journal-title":"SIAM J. Imag. Sci."},{"issue":"1","key":"2835_CR16","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1080\/00401706.1970.10488634","volume":"12","author":"AE Hoerl","year":"1970","unstructured":"Hoerl, A.E., Kennard, R.W.: Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1), 55\u201367 (1970)","journal-title":"Technometrics"},{"issue":"2","key":"2835_CR17","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1137\/070690274","volume":"51","author":"SJ Kim","year":"2009","unstructured":"Kim, S.J., Koh, K., Boyd, S., Gorinevsky, D.: $$\\ell _1$$ trend filtering. SIAM Rev. 51(2), 339\u2013360 (2009)","journal-title":"SIAM Rev."},{"issue":"4","key":"2835_CR18","first-page":"747","volume":"12","author":"GM Korpelevich","year":"1976","unstructured":"Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12(4), 747\u2013756 (1976)","journal-title":"Ekonomika i Matematicheskie Metody"},{"key":"2835_CR19","unstructured":"Li, B., Shi, B.: Understanding the ADMM algorithm via high-resolution differential equations. arXiv preprint arXiv:2401.07096 (2024)"},{"key":"2835_CR20","unstructured":"Li, B., Shi, B.: Understanding the PDHG algorithm via high-resolution differential equations. arXiv preprint arXiv:2403.11139 (2024)"},{"key":"2835_CR21","doi-asserted-by":"crossref","unstructured":"Li, B., Shi, B., Yuan, Y.X.: Proximal subgradient norm minimization of ISTA and FISTA. arXiv preprint arXiv:2211.01610 (2022)","DOI":"10.2139\/ssrn.4295674"},{"issue":"2","key":"2835_CR22","doi-asserted-by":"publisher","first-page":"2150","DOI":"10.1137\/23M158111X","volume":"34","author":"B Li","year":"2024","unstructured":"Li, B., Shi, B., Yuan, Y.X.: Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity. SIAM J. Optim. 34(2), 2150\u20132168 (2024)","journal-title":"SIAM J. Optim."},{"key":"2835_CR23","doi-asserted-by":"crossref","unstructured":"Li, B., Shi, B., Yuan, Y.X.: Linear convergence of ISTA and FISTA. Journal of the Operations Research Society of China pp. 1\u201319 (2024)","DOI":"10.1007\/s40305-024-00561-0"},{"key":"2835_CR24","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/s10444-011-9243-y","volume":"38","author":"CA Micchelli","year":"2013","unstructured":"Micchelli, C.A., Shen, L., Xu, Y., Zeng, X.: Proximity algorithms for the L1\/TV image denoising model. Adv. Comput. Math. 38, 401\u2013426 (2013)","journal-title":"Adv. Comput. Math."},{"key":"2835_CR25","unstructured":"Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol.\u00a087. Springer Science & Business Media (2013)"},{"key":"2835_CR26","first-page":"845","volume":"28","author":"LD Popov","year":"1980","unstructured":"Popov, L.D.: A modification of the Arrow-Hurwicz method for search of saddle points. Mathematical Notes of the Academy of Sciences of the USSR 28, 845\u2013848 (1980)","journal-title":"Mathematical Notes of the Academy of Sciences of the USSR"},{"key":"2835_CR27","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)","DOI":"10.1515\/9781400873173"},{"key":"2835_CR28","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer (1998)","DOI":"10.1007\/978-3-642-02431-3"},{"issue":"1\u20134","key":"2835_CR29","doi-asserted-by":"publisher","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":"2835_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-021-01681-8","volume":"195","author":"B Shi","year":"2022","unstructured":"Shi, B., Du, S.S., Jordan, M.I., Su, W.: Understanding the acceleration phenomenon via high-resolution differential equations. Math. Program. 195, 1\u201370 (2022)","journal-title":"Math. Program."},{"key":"2835_CR31","unstructured":"Shi, B., Du, S.S., Su, W., Jordan, M.I.: Acceleration via symplectic discretization of high-resolution differential equations. In: Advances in Neural Information Processing Systems, vol.\u00a032 (2019)"},{"issue":"1","key":"2835_CR32","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B Stat Methodol. 58(1), 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"issue":"1","key":"2835_CR33","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. J. R. Stat. Soc. Ser. B Stat Methodol. 67(1), 91\u2013108 (2005)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"key":"2835_CR34","doi-asserted-by":"crossref","unstructured":"Tibshirani, R.J.: The solution path of the generalized Lasso. Ph.D. thesis, Stanford University (2011)","DOI":"10.1214\/11-AOS878"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02835-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02835-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02835-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T07:44:53Z","timestamp":1758008693000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02835-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,30]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["2835"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02835-w","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2025,8,30]]},"assertion":[{"value":"23 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"67"}}