{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T03:10:49Z","timestamp":1792638649123,"version":"4.5.0"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR GOTMI"],"award-info":[{"award-number":["ANR GOTMI"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-16-CE33-0010-01"],"award-info":[{"award-number":["ANR-16-CE33-0010-01"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper we are interested in the differential inclusion $0\\in \\ddot{x}(t)+\\frac{b}{t}\\dot{x}(t)+\\partial F(x(t))$ in a finite-dimensional Hilbert space $\\mathbb{R}^{d}$, where $F$ is a proper, convex, lower semicontinuous function. The motivation of this study is that the differential inclusion models the FISTA algorithm as considered in [A. Chambolle and C. Dossal, J. Optim. Theory Appl., 166 (2015), pp. 968--982]. In particular, we investigate the different asymptotic properties of solutions for this inclusion for $b&gt;0$. We show that the convergence rate of $F(x(t))$ towards the minimum of $F$ is of order of $O\\mathopen{}(t^{-\\frac{2b}{3}})$ when $0&lt;b&lt;3$, while for $b&gt;3$ this order is of $o\\mathopen{}({t^{-2}})$ and the solution-trajectory converges to a minimizer of $F$. These results generalize the ones obtained in the differential setting (where $F$ is differentiable) in [H. Attouch, Z. Chbani, J. Peypouquet, and P. Redont, Math. Program., 2016, pp. 1--53], [H. Attouch, Z. Chbani, and H. Riahi, arXiv:1706.05671, 2017], [J. Aujol and C. Dossal, Optimal Rate of Convergence of an ODE Associated to the Fast Gradient Descent Schemes for $b&gt; 0$, 2017], and [W. Su, S. Boyd, and E. J. Candes, J. Mach. Learn. Res., 17 (2016), pp. 1--43]. In addition, we show that the order of the convergence rate $O\\mathopen{}(t^{-\\frac{2b}{3}})$ of $F(x(t))$ towards the minimum is optimal, in the case of low friction $b&lt;3$, by making a particular choice of $F$.<\/jats:p>","DOI":"10.1137\/17m1128642","type":"journal-article","created":{"date-parts":[[2018,2,22]],"date-time":"2018-02-22T14:02:31Z","timestamp":1519308151000},"page":"551-574","source":"Crossref","is-referenced-by-count":56,"title":["The Differential Inclusion Modeling FISTA Algorithm and Optimality of Convergence Rate in the Case b $\\leq3$"],"prefix":"10.1137","volume":"28","author":[{"given":"Apidopoulos","family":"Vassilis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aujol","family":"Jean-Fran\u00e7ois","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dossal","family":"Charles","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,2,22]]},"reference":[{"key":"atypb1","unstructured":"R. A. Adams and J. J. Fournier,\n                      Sobolev Spaces\n                      , Vol. 140, Academic Press, 2003."},{"key":"atypb2","first-page":"289","author":"Adly S.","year":"2006","journal-title":"Springer"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998335802"},{"key":"atypb4","unstructured":"V. Apidopoulos, J.F. Aujol, and C. Dossal,\n                      Convergence Rate of Inertial Forward-Backward Algorithm Beyond Nesterov's Rule\n                      ,https:\/\/hal.archives-ouvertes.fr\/hal-01551873(2017)."},{"key":"atypb5","first-page":"273","volume":"12","author":"Attouch H.","year":"2002","journal-title":"Adv. Math. Sci. Appl."},{"key":"atypb6","first-page":"1","author":"Attouch H.","year":"2016","journal-title":"Math. Program."},{"key":"atypb7","unstructured":"H. Attouch, Z. Chbani, and H. Riahi,\n                      Rate of Convergence of the Nesterov Accelerated Gradient Method in the Subcritical Case $\\alpha\\leq3$\n                      , preprint,arXiv:1706.05671, 2017."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1142\/S0219199700000025"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/15M1046095"},{"key":"atypb10","unstructured":"J. Aujol and C. Dossal,\n                      Optimal Rate of Convergence of an ODE Associated to the Fast Gradient Descent Schemes for $b> 0$\n                      ,https:\/\/hal.inria.fr\/hal-01547251v2(2017)."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/140994964"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"H. H. Bauschke and P. L. Combettes,\n                      Convex Analysis and Monotone Operator Theory in Hilbert Spaces\n                      , Springer Science & Business Media, 2011.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb14","unstructured":"H. Brezis,\n                      Ope\u0301rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert\n                      , Vol. 5, Elsevier, 1973."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04785-0"},{"key":"atypb16","first-page":"33","volume":"17","author":"Cabot A.","year":"2009","journal-title":"Electron. J. Differential Equations"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2007.01.003"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0746-4"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"F. Clarke,\n                      Functional Analysis, Calculus of Variations and Optimal Control\n                      , Vol. 264, Springer Science & Business Media, 2013.","DOI":"10.1007\/978-1-4471-4820-3"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"L. C. Evans and R. F. Gariepy,\n                      Measure Theory and Fine Properties of Functions\n                      , CRC Press, 2015.","DOI":"10.1201\/b18333"},{"key":"atypb21","unstructured":"N. Gigli, L. Ambrosio, G. Savare, et al.\n                      Gradient flows: In metric spaces and in the space of probability measures\n                      , Lect. Notes Math. (2005), pp. 1-333."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/0802032"},{"key":"atypb23","unstructured":"J. Heinonen,\n                      Lectures on Lipschitz Analysis\n                      , University of Jyva\u0308skyla\u0308, 2005."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2014.903569"},{"key":"atypb25","unstructured":"R. May,\n                      Asymptotic for a Second Order Evolution Equation with Convex Potential and Vanishing Damping Term\n                      , preprint,arXiv:1509.05598, 2015."},{"key":"atypb26","first-page":"372","volume":"27","author":"Nesterov Y.","year":"1983","journal-title":"Soviet Math. Dokl."},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"Y. Nesterov,\n                      Introductory Lectures on Convex Optimization: A Basic Course\n                      , Appl. Optim. 87, Kluwer Academic Publishers, Boston, MA, 2004.","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11761-0"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202500000422"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(78)90022-6"},{"key":"atypb31","first-page":"1","volume":"17","author":"Su W.","year":"2016","journal-title":"J. Mach. Learn. Res."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1128642","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:15:32Z","timestamp":1787336132000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1128642"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1128642"],"URL":"https:\/\/doi.org\/10.1137\/17m1128642","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}