{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:38:50Z","timestamp":1787341130846,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100002367","name":"Chinese Academy of Sciences","doi-asserted-by":"publisher","award":["YSBR-034"],"award-info":[{"award-number":["YSBR-034"]}],"id":[{"id":"10.13039\/501100002367","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12288201"],"award-info":[{"award-number":["12288201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>A significant milestone in modern gradient-based optimization was achieved with the development of Nesterov\u2019s accelerated gradient descent (NAG) method. This forward-backward technique has been further advanced with the introduction of its proximal generalization, commonly known as the fast iterative shrinkage-thresholding algorithm (FISTA), which enjoys widespread application in image science and engineering. Nonetheless, it remains unclear whether both NAG and FISTA exhibit linear convergence for strongly convex functions. Remarkably, these algorithms demonstrate convergence without requiring any prior knowledge of strongly convex modulus, and this intriguing characteristic has been acknowledged as an open problem in the comprehensive review [A. Chambolle and T. Pock, Acta Numer., 25 (2016), pp. 161\u2013319]. In this paper, we address this question by utilizing the high-resolution ordinary differential equation (ODE) framework. Expanding upon the established phase-space representation, we emphasize the distinctive approach employed in crafting the Lyapunov function, which involves a dynamically adapting coefficient of kinetic energy that evolves throughout the iterations. Furthermore, we highlight that the linear convergence of both NAG and FISTA is independent of the parameter [Formula: see text]. Additionally, we demonstrate that the square of the proximal subgradient norm likewise advances toward linear convergence.<\/jats:p>","DOI":"10.1137\/23m158111x","type":"journal-article","created":{"date-parts":[[2024,6,19]],"date-time":"2024-06-19T04:01:18Z","timestamp":1718769678000},"page":"2150-2168","source":"Crossref","is-referenced-by-count":9,"title":["Linear Convergence of Forward-Backward Accelerated Algorithms without Knowledge of the Modulus of Strong Convexity"],"prefix":"10.1137","volume":"34","author":[{"given":"Bowen","family":"Li","sequence":"first","affiliation":[{"name":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8616-6180","authenticated-orcid":true,"given":"Bin","family":"Shi","sequence":"additional","affiliation":[{"name":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ya-xiang","family":"Yuan","sequence":"additional","affiliation":[{"name":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,6,19]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.7153\/dea-04-04"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/15M1046095"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/130910294"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2016.08.020"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0746-4"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S096249291600009X"},{"key":"ref8","unstructured":"S. Chen, B. Shi, and Y.X. Yuan, Gradient Norm Minimization of Nesterov Acceleration, \\( o (1\/k^3)\\), preprint, arXiv:2209.08862, 2022."},{"key":"ref9","unstructured":"S. Chen, B. Shi, and Y.X. Yuan, Revisiting the High-Resolution Phenomenon via High-Resolution Differential Equations, preprint, arXiv:2212.05700, 2022."},{"key":"ref10","unstructured":"S. Chen, B. Shi, and Y.x. Yuan, On Underdamped Nesterov\u2019s Acceleration, preprint, arXiv:2304.14642, 2023."},{"key":"ref11","first-page":"295","volume":"137","author":"Gelfand I. M.","year":"1961","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref12","unstructured":"B. Li, B. Shi, and Y.X. Yuan, Linear Convergence of ISTA and FISTA, preprint, arXiv:2212.06319, 2022."},{"key":"ref13","doi-asserted-by":"crossref","unstructured":"B. Li, B. Shi, and Y.X. Yuan, Proximal Subgradient Norm Minimization of ISTA and FISTA, preprint, arXiv:2211.01610, 2022.","DOI":"10.2139\/ssrn.4295674"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01713-3"},{"key":"ref15","unstructured":"M. Muehlebach and M. Jordan, A dynamical systems perspective on Nesterov acceleration, in International Conference on Machine Learning, PMLR, 2019, pp. 4656\u20134662."},{"key":"ref16","series-title":"Appl. Optim. 87","volume-title":"Introductory Lectures on Convex Optimization: A Basic Course","author":"Nesterov Y.","year":"1998"},{"key":"ref17","first-page":"543","volume":"269","author":"Nesterov Y. E.","year":"1983","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01681-8"},{"key":"ref19","first-page":"5744","volume":"32","author":"Shi B.","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref20","first-page":"5312","volume":"17","author":"Su W.","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1614734113"},{"key":"ref22","first-page":"5040","volume":"22","author":"Wilson A. C.","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref23","first-page":"18646","volume":"34","author":"Xia H.","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref24","unstructured":"P. Zhang, A. Orvieto, H. Daneshmand, T. Hofmann, and R. S. Smith, Revisiting the role of Euler numerical integration on acceleration and stability in convex optimization, in International Conference on Artificial Intelligence and Statistics, PMLR, 2021, pp. 3979\u20133987."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M158111X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:06:59Z","timestamp":1787339219000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M158111X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,19]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M158111X"],"URL":"https:\/\/doi.org\/10.1137\/23m158111x","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,19]]}}}