{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T05:03:46Z","timestamp":1784869426297,"version":"3.55.0"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s10957-026-03052-9","type":"journal-article","created":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T05:46:22Z","timestamp":1783143982000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inertial Dynamical Systems and Accelerated Algorithms with Implicit Hessian-Driven Damping for Nonconvex Optimization"],"prefix":"10.1007","volume":"210","author":[{"given":"Zeying","family":"Gao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangkai","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liang","family":"He","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,4]]},"reference":[{"key":"3052_CR1","doi-asserted-by":"publisher","first-page":"1687","DOI":"10.1007\/s00245-020-09692-1","volume":"84","author":"CD Alecsa","year":"2021","unstructured":"Alecsa, C.D., L\u00e1szl\u00f3, S.C., Pin\u0163a, T.: An extension of the second-order dynamical system that models Nesterov\u2019s convex gradient method. Appl. Math. Optim. 84, 1687\u20131716 (2021)","journal-title":"Appl. Math. Optim."},{"key":"3052_CR2","doi-asserted-by":"publisher","first-page":"2921","DOI":"10.1137\/20M1382027","volume":"31","author":"CD Alecsa","year":"2021","unstructured":"Alecsa, C.D., L\u00e1szl\u00f3, S.C.: Tikhonov regularization of a perturbed heavy ball system with vanishing damping. SIAM J. Optim. 31, 2921\u20132954 (2021)","journal-title":"SIAM J. Optim."},{"key":"3052_CR3","doi-asserted-by":"crossref","unstructured":"Alvarez, 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, 747\u2013779 (2002)","DOI":"10.1016\/S0021-7824(01)01253-3"},{"key":"3052_CR4","doi-asserted-by":"publisher","first-page":"1985","DOI":"10.4171\/jems\/1231","volume":"25","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Bo\u0163, R.I., Csetnek, E.R.: Fast optimization via inertial dynamics with closedloop damping. J. Eur. Math. Soc. 25, 1985\u20132056 (2023)","journal-title":"J. Eur. Math. Soc."},{"key":"3052_CR5","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."},{"key":"3052_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 The. 7, 353\u2013371 (2018)","journal-title":"Evol. Equ. Control The."},{"key":"3052_CR7","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10107-020-01591-1","volume":"193","author":"H Attouch","year":"2022","unstructured":"Attouch, H., Chbani, Z., Fadili, J., Riahi, H.: First-order optimization algorithms via inertial systems with Hessian-driven damping. Math. Program. 193, 113\u2013155 (2022)","journal-title":"Math. Program."},{"key":"3052_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. 168, 123\u2013175 (2018)","journal-title":"Math. Program."},{"key":"3052_CR9","doi-asserted-by":"publisher","first-page":"1065","DOI":"10.1016\/j.jmaa.2016.12.017","volume":"457","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Chbani, Z., Riahi, H.: Combining fast inertial dynamics for convex optimization with Tikhonov regularization. J. Math. Anal. Appl. 457, 1065\u20131094 (2018)","journal-title":"J. Math. Anal. Appl."},{"key":"3052_CR10","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 (2019)","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"3052_CR11","doi-asserted-by":"publisher","first-page":"71","DOI":"10.3934\/eect.2022022","volume":"12","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Fadili, J., Kungurtsev, V.: On the effect of perturbations in first-order optimization methods with inertia and Hessian-driven damping. Evol. Equ. Control Theory. 12, 71\u2013117 (2023)","journal-title":"Evol. Equ. Control Theory."},{"key":"3052_CR12","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-backward algorithm for convex minimization. SIAM J. Optim. 24, 232\u2013256 (2014)","journal-title":"SIAM J. Optim."},{"key":"3052_CR13","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 optimization via inertial dynamics with Hessian-driven damping. J. Differ. Equ. 261, 5734\u20135783 (2016)","journal-title":"J. Differ. Equ."},{"key":"3052_CR14","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/s00245-023-10058-6","volume":"88","author":"JF Aujol","year":"2023","unstructured":"Aujol, J.F., Dossal, C., Ho\u00e0ng, V.H., Labarri\u00e8re, H., Rondepierre, A.: Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions. Appl. Math. Optim. 88, 81 (2023)","journal-title":"Appl. Math. Optim."},{"key":"3052_CR15","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10107-020-01528-8","volume":"189","author":"RI Bot","year":"2021","unstructured":"Bot, R.I., Csetnek, E.R., L\u00e1szl\u00f3, S.C.: Tikhonov regularization of a second-order dynamical system with Hessian-driven damping. Math. Program. 189, 151\u2013186 (2021)","journal-title":"Math. Program."},{"key":"3052_CR16","unstructured":"Bot, R.I., Fadili, J., Nguyen, D.K.: The iterates of Nesterov\u2019s accelerated algorithm converge in the critical regimes. arXiv: 2510.22715v2 (2025)"},{"key":"3052_CR17","volume-title":"Generalized Convexity and Optimization: Theory and Applications","author":"A Cambini","year":"2009","unstructured":"Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Springer, Berlin (2009)"},{"key":"3052_CR18","first-page":"1","volume":"22","author":"C Castera","year":"2021","unstructured":"Castera, C., Bolte, J., F\u00e9votte, C., Pauwels, E.: An inertial Newton algorithm for deep learning. J. Mach. Learn. Res. 22, 1\u201331 (2021)","journal-title":"J. Mach. Learn. Res."},{"key":"3052_CR19","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1007\/s10957-016-0938-6","volume":"170","author":"F Flores-Baz\u00e1n","year":"2016","unstructured":"Flores-Baz\u00e1n, F., Hadjisavvas, N., Lara, F., Montenegro, I.: First- and second-order asymptotic analysis with applications in quasiconvex optimization. J. Optim. Theory Appl. 170, 372\u2013393 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR20","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s10107-007-0109-5","volume":"116","author":"X Goudou","year":"2009","unstructured":"Goudou, X., Munier, J.: The gradient and Heavy Ball with friction dynamical systems: the quasiconvex case. Math. Program. 116, 173\u2013191 (2009)","journal-title":"Math. Program."},{"key":"3052_CR21","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s10898-022-01226-z","volume":"85","author":"S-M Grad","year":"2023","unstructured":"Grad, S.-M., Lara, F., Marcavillaca, R.T.: Relaxed-inertial proximal point type algorithms for quasiconvex minimization. J. Glob. Optim. 85, 615\u2013635 (2023)","journal-title":"J. Glob. Optim."},{"key":"3052_CR22","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/s10957-025-02641-4","volume":"205","author":"S-M Grad","year":"2025","unstructured":"Grad, S.-M., Lara, F., Marcavillaca, R.T.: Strongly quasiconvex functions: what we know (so far). J. Optim. Theory Appl. 205, 38 (2025)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR23","volume-title":"Handbook of Generalized Convexity and Generalized Monotonicity","author":"N Hadjisavvas","year":"2006","unstructured":"Hadjisavvas, N., Koml\u00f3si, S., Schaible, S.S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, Boston (2006)"},{"key":"3052_CR24","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s00245-026-10406-2","volume":"93","author":"N Hadjisavvas","year":"2026","unstructured":"Hadjisavvas, N., Lara, F., Marcavillaca, R.T., Vuong, P.T.: Heavy ball and Nesterov accelerations with Hessian-driven damping for nonconvex optimization. Appl. Math. Optim. 93, 63 (2026)","journal-title":"Appl. Math. Optim."},{"key":"3052_CR25","doi-asserted-by":"publisher","first-page":"170","DOI":"10.1007\/s10957-018-1317-2","volume":"180","author":"N Hadjisavvas","year":"2019","unstructured":"Hadjisavvas, N., Lara, F., Mart\u00ednez-Legaz, J.E.: A quasiconvex asymptotic function with applications in optimization. J. Optim. Theory Appl. 180, 170\u2013186 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR26","doi-asserted-by":"crossref","unstructured":"He, X., Fang, Y.P., Accelerated forward-backward algorithms with subgradient corrections. Comput. Optim. Appl. 93, 121156 (2026)","DOI":"10.1007\/s10589-025-00719-3"},{"key":"3052_CR27","unstructured":"Hinder, O., Sidford, A., Sohoni, N.: Near-optimal methods for minimizing star-convex functions and beyond. In: Proc. 33th Conf. on Lear. Theory, vol. 125, pp. 1894\u20131938. (2020)"},{"key":"3052_CR28","unstructured":"Jang, U., Ryu, E.K.: Point convergence of Nesterov\u2019s accelerated gradient method: an AI-assisted proof. arXiv: 2510.23513v2 (2025)"},{"key":"3052_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01097545","volume":"44","author":"AI Korablev","year":"1989","unstructured":"Korablev, A.I.: Relaxation methods of minimization of pseudoconvex functions. J. Sov. Math. 44, 1\u20135 (1989)","journal-title":"J. Sov. Math."},{"key":"3052_CR30","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s10957-021-01996-8","volume":"192","author":"F Lara","year":"2022","unstructured":"Lara, F.: On strongly quasiconvex functions: existence results and proximal point algorithms. J. Optim. Theory Appl. 192, 891\u2013911 (2022)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR31","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1007\/s10957-025-02728-y","volume":"206","author":"F Lara","year":"2025","unstructured":"Lara, F., Marcavillaca, R.T., Vuong, P.T.: Characterizations, dynamical systems and gradient methods for strongly quasiconvex functions. J. Optim. Theory Appl. 206, 60 (2025)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR32","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.jde.2023.03.014","volume":"362","author":"SC L\u00e1szl\u00f3","year":"2023","unstructured":"L\u00e1szl\u00f3, S.C.: On the strong convergence of the trajectories of a Tikhonov regularized second-order dynamical system with asymptotically vanishing damping. J. Differ. Equ. 362, 355\u2013381 (2023)","journal-title":"J. Differ. Equ."},{"key":"3052_CR33","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10589-024-00620-5","volume":"90","author":"SC L\u00e1szl\u00f3","year":"2025","unstructured":"L\u00e1szl\u00f3, S.C.: Solving convex optimization problems via a second-order dynamical system with implicit Hessian damping and Tikhonov regularization. Comput. Optim. Appl. 90, 113\u2013149 (2025)","journal-title":"Comput. Optim. Appl."},{"key":"3052_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2025.109480","volume":"153","author":"XJ Long","year":"2026","unstructured":"Long, X.J., Nie, J.L., Gou, Z., Sun, X.K., Li, G.X.: An accelerated preconditioned primal-dual gradient algorithm for structured nonconvex optimization problems. Commun. Nonlinear Sci. Numer. Simulat. 153, 109480 (2026)","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"3052_CR35","doi-asserted-by":"publisher","first-page":"1867","DOI":"10.4208\/jcm.2512-m2024-0107","volume":"44","author":"XJ Long","year":"2026","unstructured":"Long, X.J., Nie, J.L., Li, G.X., Peng, Z.Y.: An accelerated preconditioned primal-dual gradient algorithm for nonconvex composite optimization problems with applications. J. Comput. Math. 44, 1867\u20131889 (2026)","journal-title":"J. Comput. Math."},{"key":"3052_CR36","unstructured":"\u0141ojasiewicz, S.: A topological property of real analytic subsets. Coll. du. CNRS, Les \u00e9quations aux d\u00e9riv\u00e9es partielles 117, 87-89 (1963)"},{"key":"3052_CR37","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1007\/s11590-025-02261-x","volume":"20","author":"NM Nam","year":"2026","unstructured":"Nam, N.M., Sharkansky, J.: On strong quasiconvexity of functions in infinite dimensions. Optim. Lett. 20, 815\u2013840 (2026)","journal-title":"Optim. Lett."},{"key":"3052_CR38","first-page":"543","volume":"269","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method for solving the convex programming problem with convergence rate $$\\cal{O} (1\/k^2)$$. Dokl. Akad. Nauk SSSR 269, 543\u2013547 (1983)","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"3052_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8853-9","volume-title":"Introductory Lectures on Convex Optimization","author":"Y Nesterov","year":"2004","unstructured":"Nesterov, Y.: Introductory Lectures on Convex Optimization. Springer, New York (2004)"},{"key":"3052_CR40","first-page":"643","volume":"3","author":"BT Polyak","year":"1963","unstructured":"Polyak, B.T.: Gradient methods for minimizing functionals. Zh. Vychisl. Math. Mat. Fiz. 3, 643\u2013653 (1963)","journal-title":"Zh. Vychisl. Math. Mat. Fiz."},{"key":"3052_CR41","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, 1\u201317 (1964)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"3052_CR42","first-page":"72","volume":"7","author":"BT Polyak","year":"1966","unstructured":"Polyak, B.T.: Existence theorem and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. 7, 72\u201375 (1966)","journal-title":"Soviet Math."},{"key":"3052_CR43","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s10957-020-01784-w","volume":"188","author":"M Rahimi Piranfar","year":"2021","unstructured":"Rahimi Piranfar, M., Khatibzadeh, H.: Long-time behavior of a gradient system governed by a quasiconvex function. J. Optim. Theory Appl. 188, 169\u2013191 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"3052_CR44","first-page":"1","volume":"17","author":"W Su","year":"2016","unstructured":"Su, W., Boyd, S., Cand\u00e8s, E.J.: A differential equation for modeling Nesterov\u2019s accelerated gradient method: theory and insights. J. Mach. Learn. Res. 17, 1\u201343 (2016)","journal-title":"J. Mach. Learn. Res."},{"key":"3052_CR45","first-page":"18","volume":"4","author":"AA Vladimirov","year":"1978","unstructured":"Vladimirov, A.A., Nesterov, Y.E., Chekanov, Y.N.: On uniformly quasi-convex functionals. Vestn. Mosk. Univ. 4, 18\u201327 (1978)","journal-title":"Vestn. Mosk. Univ."},{"key":"3052_CR46","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-025-01540-2","author":"T Wang","year":"2025","unstructured":"Wang, T., Cai, X.J., Li, T.: On the $$o(1\/k^2)$$ convergence rate analysis for general inertial gradient type method. J. Glob. Optim. (2025). https:\/\/doi.org\/10.1007\/s10898-025-01540-2","journal-title":"J. Glob. Optim."},{"key":"3052_CR47","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1007\/s10957-024-02462-x","volume":"203","author":"G Zhong","year":"2024","unstructured":"Zhong, G., Hu, X., Tang, M., Zhong, L.: Fast convex optimization via differential equation with Hessian-driven damping and Tikhonov regularization. J. Optim. Theory Appl. 203, 42\u201382 (2024)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-03052-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-026-03052-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-03052-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T04:51:16Z","timestamp":1784868676000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-026-03052-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["3052"],"URL":"https:\/\/doi.org\/10.1007\/s10957-026-03052-9","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7]]},"assertion":[{"value":"7 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"No potential conflict of interest was reported by the authors.","order":1,"name":"Ethics","label":"Conflicts of Interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"21"}}