{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T23:37:06Z","timestamp":1780702626237,"version":"3.54.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T00:00:00Z","timestamp":1582070400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T00:00:00Z","timestamp":1582070400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s10107-020-01476-3","type":"journal-article","created":{"date-parts":[[2020,2,19]],"date-time":"2020-02-19T16:03:08Z","timestamp":1582128188000},"page":"151-193","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Convergence rates of an inertial gradient descent algorithm under growth and flatness conditions"],"prefix":"10.1007","volume":"187","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9295-5431","authenticated-orcid":false,"given":"Vassilis","family":"Apidopoulos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Fran\u00e7ois","family":"Aujol","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Charles","family":"Dossal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aude","family":"Rondepierre","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,2,19]]},"reference":[{"key":"1476_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-018-1350-9","author":"V Apidopoulos","year":"2018","unstructured":"Apidopoulos, V., Aujol, J.F., Dossal, C.: Convergence rate of inertial forward-backward algorithm beyond Nesterov\u2019s rule. Math. Program. (2018). https:\/\/doi.org\/10.1007\/s10107-018-1350-9","journal-title":"Math. Program."},{"issue":"1","key":"1476_CR2","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1137\/17M1128642","volume":"28","author":"V Apidopoulos","year":"2018","unstructured":"Apidopoulos, V., Aujol, J.F., Dossal, C.: The differential inclusion modeling FISTA algorithm and optimality of convergence rate in the case b $$\\le 3$$. SIAM J. Optim. 28(1), 551\u2013574 (2018)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1476_CR3","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. 116(1), 5\u201316 (2009)","journal-title":"Math. Program."},{"issue":"2","key":"1476_CR4","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the kurdyka-\u0142ojasiewicz inequality. Math. Oper. Res. 35(2), 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"1476_CR5","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1137\/17M1114739","volume":"28","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Cabot, A.: Convergence rates of inertial forward-backward algorithms. SIAM J. Optim. 28(1), 849\u2013874 (2018)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"1476_CR6","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s10107-016-0992-8","volume":"168","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 168(1\u20132), 123\u2013175 (2018)","journal-title":"Math. Program."},{"key":"1476_CR7","unstructured":"Attouch, H., Chbani, Z., Riahi, H.: Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $$\\alpha \\le 3$$. arXiv preprint arXiv:1706.05671 (2017)"},{"issue":"3","key":"1476_CR8","doi-asserted-by":"publisher","first-page":"1824","DOI":"10.1137\/15M1046095","volume":"26","author":"H Attouch","year":"2016","unstructured":"Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov\u2019s accelerated forward-backward method is actually faster than $$1\/\\text{ k }^2$$. SIAM J. Optim. 26(3), 1824\u20131834 (2016)","journal-title":"SIAM J. Optim."},{"key":"1476_CR9","unstructured":"Aujol, J.F., Dossal, C.: Optimal rate of convergence of an ode associated to the fast gradient descent schemes for $$b>0$$. J. Differ. Equ. (2017) (submitted to)"},{"issue":"4","key":"1476_CR10","doi-asserted-by":"publisher","first-page":"3131","DOI":"10.1137\/18M1186757","volume":"29","author":"JF Aujol","year":"2019","unstructured":"Aujol, J.F., Dossal, C., Rondepierre, A.: Optimal convergence rates for Nesterov acceleration. SIAM J. Optim. 29(4), 3131\u20133153 (2019)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1476_CR11","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183\u2013202 (2009)","journal-title":"SIAM J. Imaging Sci."},{"issue":"4","key":"1476_CR12","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2006","unstructured":"Bolte, J., Daniilidis, A., Lewis, A.: The \u0141ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205\u20131223 (2006). https:\/\/doi.org\/10.1137\/050644641. (electronic)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"1476_CR13","doi-asserted-by":"publisher","first-page":"3319","DOI":"10.1090\/S0002-9947-09-05048-X","volume":"362","author":"J Bolte","year":"2010","unstructured":"Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of \u0141ojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319\u20133363 (2010)","journal-title":"Trans. Am. Math. Soc."},{"issue":"2","key":"1476_CR14","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s10107-016-1091-6","volume":"165","author":"J Bolte","year":"2017","unstructured":"Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program. 165(2), 471\u2013507 (2017)","journal-title":"Math. Program."},{"issue":"11","key":"1476_CR15","doi-asserted-by":"publisher","first-page":"5983","DOI":"10.1090\/S0002-9947-09-04785-0","volume":"361","author":"A Cabot","year":"2009","unstructured":"Cabot, A., Engler, H., Gadat, S.: On the long time behavior of second order differential equations with asymptotically small dissipation. Trans. Am. Math. Soc. 361(11), 5983\u20136017 (2009)","journal-title":"Trans. Am. Math. Soc."},{"key":"1476_CR16","unstructured":"Calatroni, L., Chambolle, A.: Backtracking strategies for accelerated descent methods with smooth composite objectives. arXiv preprint arXiv:1709.09004 (2017)"},{"issue":"3","key":"1476_CR17","doi-asserted-by":"publisher","first-page":"968","DOI":"10.1007\/s10957-015-0746-4","volume":"166","author":"A Chambolle","year":"2015","unstructured":"Chambolle, A., Dossal, C.: On the convergence of the iterates of the \u201cfast iterative shrinkage\/thresholding algorithm\u201d. J. Optim. Theory Appl. 166(3), 968\u2013982 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1476_CR18","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1287\/moor.2017.0889","volume":"43","author":"D Drusvyatskiy","year":"2018","unstructured":"Drusvyatskiy, D., Lewis, A.: Error bounds, quadratic growth, and linear convergence of proximal methods. Math. Oper. Res. 43(3), 919\u2013948 (2018)","journal-title":"Math. Oper. Res."},{"key":"1476_CR19","unstructured":"Fercoq, O., Qu, Z.: Restarting accelerated gradient methods with a rough strong convexity estimate. arXiv preprint arXiv:1609.07358 (2016)"},{"key":"1476_CR20","unstructured":"Fercoq, O., Qu, Z.: Adaptive restart of accelerated gradient methods under local quadratic growth condition. arXiv preprint arXiv:1709.02300 (2017)"},{"issue":"3","key":"1476_CR21","doi-asserted-by":"publisher","first-page":"874","DOI":"10.1007\/s10957-014-0642-3","volume":"165","author":"P Frankel","year":"2015","unstructured":"Frankel, P., Garrigos, G., Peypouquet, J.: Splitting methods with variable metric for Kurdyka-\u0142ojasiewicz functions and general convergence rates. J. Optim. Theory Appl. 165(3), 874\u2013900 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1476_CR22","unstructured":"Garrigos, G., Rosasco, L., Villa, S.: Convergence of the forward-backward algorithm: Beyond the worst case with the help of geometry. arXiv preprint arXiv:1703.09477 (2017)"},{"issue":"4","key":"1476_CR23","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1137\/0802032","volume":"2","author":"O G\u00fcler","year":"1992","unstructured":"G\u00fcler, O.: New proximal point algorithms for convex minimization. SIAM J. Optim. 2(4), 649\u2013664 (1992)","journal-title":"SIAM J. Optim."},{"key":"1476_CR24","unstructured":"Holte, J.M.: Discrete Gronwall lemma and applications. MAA-NCS meeting at the University of North Dakota 24, 1\u20137 (2009)"},{"issue":"3","key":"1476_CR25","doi-asserted-by":"publisher","first-page":"688","DOI":"10.1007\/s10957-018-1226-4","volume":"176","author":"F Iutzeler","year":"2018","unstructured":"Iutzeler, F., Malick, J.: On the proximal gradient algorithm with alternated inertia. J. Optim. Theory Appl. 176(3), 688\u2013710 (2018)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"1476_CR26","doi-asserted-by":"publisher","first-page":"705","DOI":"10.1007\/s11228-015-0330-y","volume":"23","author":"AY Kruger","year":"2015","unstructured":"Kruger, A.Y.: Error bounds and h\u00f6lder metric subregularity. Set Val. Var. Anal. 23(4), 705\u2013736 (2015)","journal-title":"Set Val. Var. Anal."},{"issue":"1","key":"1476_CR27","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1137\/15M1009597","volume":"26","author":"L Lessard","year":"2016","unstructured":"Lessard, L., Recht, B., Packard, A.: Analysis and design of optimization algorithms via integral quadratic constraints. SIAM J. Optim. 26(1), 57\u201395 (2016). https:\/\/doi.org\/10.1137\/15M1009597","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1476_CR28","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1137\/16M106340X","volume":"27","author":"J Liang","year":"2017","unstructured":"Liang, J., Fadili, J., Peyr\u00e9, G.: Activity identification and local linear convergence of forward-backward-type methods. SIAM J. Optim. 27(1), 408\u2013437 (2017)","journal-title":"SIAM J. Optim."},{"key":"1476_CR29","unstructured":"Liu, M., Yang, T.: Adaptive accelerated gradient converging method under h$$\\backslash $$\u201d$$\\{$$o$$\\}$$ lderian error bound condition. In: Advances in Neural Information Processing Systems 30 (NIPS), pp. 3106\u20133116 (2017)"},{"key":"1476_CR30","unstructured":"\u0141ojasiewicz, S.: Une propri\u00e9t\u00e9 topologique des sous-ensembles analytiques r\u00e9els. In: Les \u00c9quations aux D\u00e9riv\u00e9es Partielles (Paris, 1962), pp. 87\u201389. \u00c9ditions du Centre National de la Recherche Scientifique, Paris (1963)"},{"issue":"5","key":"1476_CR31","doi-asserted-by":"publisher","first-page":"1575","DOI":"10.5802\/aif.1384","volume":"43","author":"S \u0141ojasiewicz","year":"1993","unstructured":"\u0141ojasiewicz, S.: Sur la g\u00e9om\u00e9trie semi- et sous-analytique. Annales de l\u2019Institut Fourier 43(5), 1575\u20131595 (1993)","journal-title":"Annales de l\u2019Institut Fourier"},{"key":"1476_CR32","unstructured":"May, R.: Asymptotic for a second order evolution equation with convex potential and vanishing damping term. arXiv preprint arXiv:1509.05598 (2015)"},{"issue":"3","key":"1476_CR33","doi-asserted-by":"publisher","first-page":"685","DOI":"10.3934\/cpaa.2010.9.685","volume":"9","author":"B Merlet","year":"2010","unstructured":"Merlet, B., Pierre, M.: Convergence to equilibrium for the backward Euler scheme and applications. Commun. Pure Appl. Anal. 9(3), 685\u2013702 (2010)","journal-title":"Commun. Pure Appl. Anal."},{"key":"1476_CR34","doi-asserted-by":"crossref","unstructured":"Necoara, I., Nesterov, Y., Glineur, F.o.: Linear convergence of first order methods for non-strongly convex optimization. In: Mathematical Programming pp. 1\u201339 (2018)","DOI":"10.1007\/s10107-018-1232-1"},{"issue":"2","key":"1476_CR35","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0041-5553(85)90100-4","volume":"25","author":"AS Nemirovskii","year":"1985","unstructured":"Nemirovskii, A.S., Nesterov, Y.E.: Optimal methods of smooth convex minimization. USSR Comput. Math. Math. Phys. 25(2), 21\u2013309 (1985)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"1476_CR36","first-page":"372","volume":"27","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method of solving a convex programming problem with convergence rate $$o (1\/k^2)$$. Soviet Math. Doklady 27, 372\u2013376 (1983)","journal-title":"Soviet Math. Doklady"},{"key":"1476_CR37","unstructured":"Nesterov, Y.: Introductory lectures on convex optimization: a basic course (2013)"},{"issue":"3","key":"1476_CR38","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1007\/s10208-013-9150-3","volume":"15","author":"B O\u2019donoghue","year":"2015","unstructured":"O\u2019donoghue, B., Candes, E.: Adaptive restart for accelerated gradient schemes. Found. Comput. Math. 15(3), 715\u2013732 (2015)","journal-title":"Found. Comput. Math."},{"issue":"4","key":"1476_CR39","doi-asserted-by":"publisher","first-page":"864","DOI":"10.1016\/0041-5553(63)90382-3","volume":"3","author":"BT Polyak","year":"1963","unstructured":"Polyak, B.T.: Gradient methods for the minimisation of functionals. USSR Comput. Math. Math. Phys. 3(4), 864\u2013878 (1963)","journal-title":"USSR Comput. Math. Math. Phys."},{"issue":"5","key":"1476_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","volume":"4","author":"BT Polyak","year":"1964","unstructured":"Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1\u201317 (1964)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"1476_CR41","volume-title":"Variational analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational analysis, vol. 317. Springer, Berlin (2009)"},{"key":"1476_CR42","unstructured":"Roulet, V., d\u2019Aspremont, A.: Sharpness, restart and acceleration. In: Advances in Neural Information Processing Systems, pp. 1119\u20131129 (2017)"},{"key":"1476_CR43","unstructured":"Schmidt, M., Le\u00a0Roux, N., Bach, F.: Convergence rates of inexact proximal-gradient methods for convex optimization. In: NIPS (2011)"},{"key":"1476_CR44","unstructured":"Scieur, D., Roulet, V., Bach, F., d\u2019Aspremont, A.: Integration methods and optimization algorithms. In: I.\u00a0Guyon, U.V. Luxburg, S.\u00a0Bengio, H.\u00a0Wallach, R.\u00a0Fergus, S.\u00a0Vishwanathan, R.\u00a0Garnett (eds.) Advances in Neural Information Processing Systems 30, pp. 1109\u20131118. Curran Associates, Inc. (2017). http:\/\/papers.nips.cc\/paper\/6711-integration-methods-and-optimization-algorithms.pdf"},{"issue":"153","key":"1476_CR45","first-page":"1","volume":"17","author":"W Su","year":"2016","unstructured":"Su, W., Boyd, S., Candes, E.J.: A differential equation for modeling Nesterov\u2019s accelerated gradient method: theory and insights. J. Mach. Learn. Res. 17(153), 1\u201343 (2016)","journal-title":"J. Mach. Learn. Res."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01476-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-020-01476-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01476-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,20]],"date-time":"2021-04-20T18:46:16Z","timestamp":1618944376000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-020-01476-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,19]]},"references-count":45,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["1476"],"URL":"https:\/\/doi.org\/10.1007\/s10107-020-01476-3","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,19]]},"assertion":[{"value":"24 December 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 January 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}