{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T15:29:15Z","timestamp":1786980555846,"version":"3.56.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T00:00:00Z","timestamp":1605484800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T00:00:00Z","timestamp":1605484800000},"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":[[2022,5]]},"DOI":"10.1007\/s10107-020-01591-1","type":"journal-article","created":{"date-parts":[[2020,11,16]],"date-time":"2020-11-16T10:05:31Z","timestamp":1605521131000},"page":"113-155","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":101,"title":["First-order optimization algorithms via inertial systems with Hessian driven damping"],"prefix":"10.1007","volume":"193","author":[{"given":"Hedy","family":"Attouch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zaki","family":"Chbani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jalal","family":"Fadili","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hassan","family":"Riahi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,11,16]]},"reference":[{"issue":"4","key":"1591_CR1","doi-asserted-by":"publisher","first-page":"1102","DOI":"10.1137\/S0363012998335802","volume":"38","author":"F \u00c1lvarez","year":"2000","unstructured":"\u00c1lvarez, F.: On the minimizing property of a second-order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38(4), 1102\u20131119 (2000)","journal-title":"SIAM J. Control Optim."},{"issue":"8","key":"1591_CR2","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1016\/S0021-7824(01)01253-3","volume":"81","author":"F \u00c1lvarez","year":"2002","unstructured":"\u00c1lvarez, F., Attouch, H., Bolte, J., Redont, P.: A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. J. Math. Pures Appl. 81(8), 747\u2013779 (2002)","journal-title":"J. Math. Pures Appl."},{"key":"1591_CR3","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s10107-018-1350-9","volume":"180","author":"V Apidopoulos","year":"2020","unstructured":"Apidopoulos, V., Aujol, J.-F., Dossal, C.: Convergence rate of inertial Forward\u2013Backward algorithm beyond Nesterov\u2019s rule. Math. Program. Ser. B. 180, 137\u2013156 (2020)","journal-title":"Math. Program. Ser. B."},{"key":"1591_CR4","doi-asserted-by":"publisher","first-page":"5412","DOI":"10.1016\/j.jde.2017.06.024","volume":"263","author":"H Attouch","year":"2017","unstructured":"Attouch, H., Cabot, A.: Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity. J. Differ. Equ. 263, 5412\u20135458 (2017)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"1591_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\u2013backward algorithms. SIAM J. Optim. 28(1), 849\u2013874 (2018)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"1591_CR6","doi-asserted-by":"publisher","first-page":"353","DOI":"10.3934\/eect.2018018","volume":"7","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Cabot, A., Chbani, Z., Riahi, H.: Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient. Evol. Equ. Control Theory 7(3), 353\u2013371 (2018)","journal-title":"Evol. Equ. Control Theory"},{"issue":"3","key":"1591_CR7","doi-asserted-by":"publisher","first-page":"2227","DOI":"10.1137\/18M1230207","volume":"29","author":"H Attouch","year":"2019","unstructured":"Attouch, H., Chbani, Z., Riahi, H.: Fast proximal methods via time scaling of damped inertial dynamics. SIAM J. Optim. 29(3), 2227\u20132256 (2019)","journal-title":"SIAM J. Optim."},{"key":"1591_CR8","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. Ser. B. 168, 123\u2013175 (2018)","journal-title":"Math. Program. Ser. B."},{"key":"1591_CR9","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1051\/cocv\/2017083","volume":"25","author":"H Attouch","year":"2019","unstructured":"Attouch, H., Chbani, Z., Riahi, H.: Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $$\\alpha \\le 3$$. ESAIM Control Optim. Calc. Var. 25, 2\u201335 (2019)","journal-title":"ESAIM Control Optim. Calc. Var."},{"issue":"3","key":"1591_CR10","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\u2013backward method is actually faster than $$1\/k^2$$. SIAM J. Optim. 26(3), 1824\u20131834 (2016)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1591_CR11","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1137\/130910294","volume":"24","author":"H Attouch","year":"2014","unstructured":"Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward\u2013backward algorithm for convex minimization. SIAM J. Optim. 24(1), 232\u2013256 (2014)","journal-title":"SIAM J. Optim."},{"issue":"10","key":"1591_CR12","doi-asserted-by":"publisher","first-page":"5734","DOI":"10.1016\/j.jde.2016.08.020","volume":"261","author":"H Attouch","year":"2016","unstructured":"Attouch, H., Peypouquet, J., Redont, P.: Fast convex minimization via inertial dynamics with Hessian driven damping. J. Diffe. Equ. 261(10), 5734\u20135783 (2016)","journal-title":"J. Diffe. Equ."},{"key":"1591_CR13","doi-asserted-by":"crossref","unstructured":"Attouch, H., Svaiter, B. F.: A continuous dynamical Newton-Like approach to solving monotone inclusions. SIAM J. Control Optim. 49(2), 574\u2013598 (2011). Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces. J. Optim. Theory Appl. 157(3), 624\u2013650 (2013)","DOI":"10.1137\/100784114"},{"issue":"4","key":"1591_CR14","doi-asserted-by":"publisher","first-page":"2408","DOI":"10.1137\/140994964","volume":"25","author":"J-F Aujol","year":"2015","unstructured":"Aujol, J.-F., Dossal, Ch.: Stability of over-relaxations for the forward-backward algorithm, application to FISTA. SIAM J. Optim. 25(4), 2408\u20132433 (2015)","journal-title":"SIAM J. Optim."},{"key":"1591_CR15","unstructured":"Aujol, J.-F., Dossal, C.: Optimal rate of convergence of an ODE associated to the Fast Gradient Descent schemes for $$b>0$$ (2017). https:\/\/hal.inria.fr\/hal-01547251v2"},{"key":"1591_CR16","volume-title":"Higher Transcendental Functions","author":"H Bateman","year":"1953","unstructured":"Bateman, H.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)"},{"key":"1591_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"H Bauschke","year":"2011","unstructured":"Bauschke, H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics, Springer (2011)"},{"issue":"1","key":"1591_CR18","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."},{"key":"1591_CR19","unstructured":"Br\u00e9zis, H.: Op\u00e9rateurs maximaux monotones dans les espaces de Hilbert et \u00e9quations d\u2019\u00e9volution, Lecture Notes 5, North Holland, (1972)"},{"key":"1591_CR20","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, 5983\u20136017 (2009)","journal-title":"Trans. Am. Math. Soc."},{"key":"1591_CR21","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, Ch.: On the convergence of the iterates of the fast iterative shrinkage thresholding algorithm. J. Optim. Theory Appl. 166, 968\u2013982 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1591_CR22","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."},{"key":"1591_CR23","unstructured":"Gelfand, I.M., Zejtlin, M.: Printszip nelokalnogo poiska v sistemah avtomatich, Optimizatsii, Dokl. AN SSSR, 137, 295?298 (1961) (in Russian)"},{"issue":"3","key":"1591_CR24","doi-asserted-by":"publisher","first-page":"681","DOI":"10.3906\/mat-1512-28","volume":"41","author":"R May","year":"2017","unstructured":"May, R.: Asymptotic for a second-order evolution equation with convex potential and vanishing damping term. Turk. J. Math. 41(3), 681\u2013685 (2017)","journal-title":"Turk. J. Math."},{"key":"1591_CR25","first-page":"372","volume":"27","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method of solving a convex programming problem with convergence rate $$O(1\/k^2)$$. Sov. Math. Doklady 27, 372\u2013376 (1983)","journal-title":"Sov. Math. Doklady"},{"issue":"1\u20132","key":"1591_CR26","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s10107-014-0790-0","volume":"152","author":"Y Nesterov","year":"2015","unstructured":"Nesterov, Y.: Gradient methods for minimizing composite objective function. Math. Program. 152(1\u20132), 381\u2013404 (2015)","journal-title":"Math. Program."},{"key":"1591_CR27","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. U.S.S.R. Comput. Math. Math. Phys. 4, 1\u201317 (1964)","journal-title":"U.S.S.R. Comput. Math. Math. Phys."},{"key":"1591_CR28","volume-title":"Introduction to Optimization","author":"BT Polyak","year":"1987","unstructured":"Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)"},{"key":"1591_CR29","unstructured":"Siegel, W.: Accelerated first-order methods: differential equations and Lyapunov functions. arXiv:1903.05671v1 [math.OC] (2019)"},{"key":"1591_CR30","unstructured":"Shi, B., Du, S.S., Jordan, M.I., Su, W.J.: Understanding the acceleration phenomenon via high-resolution differential equations. arXiv:submit\/2440124 [cs.LG] 21 Oct 2018"},{"key":"1591_CR31","first-page":"2510","volume":"27","author":"WJ Su","year":"2014","unstructured":"Su, W.J., Boyd, S., Cand\u00e8s, E.J.: A differential equation for modeling Nesterov\u2019s accelerated gradient method: theory and insights. NIPS\u201914 27, 2510\u20132518 (2014)","journal-title":"NIPS\u201914"},{"key":"1591_CR32","unstructured":"Wilson, A.C., Recht, B., Jordan, M.I.: A Lyapunov analysis of momentum methods in optimization. arXiv:1611.02635 (2016)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01591-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-020-01591-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01591-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,25]],"date-time":"2022-04-25T14:09:06Z","timestamp":1650895746000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-020-01591-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,16]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,5]]}},"alternative-id":["1591"],"URL":"https:\/\/doi.org\/10.1007\/s10107-020-01591-1","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,16]]},"assertion":[{"value":"24 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 November 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}