{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T23:09:47Z","timestamp":1782169787073,"version":"3.54.5"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,5,6]],"date-time":"2022-05-06T00:00:00Z","timestamp":1651795200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,5,6]],"date-time":"2022-05-06T00:00:00Z","timestamp":1651795200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study convergence of the trajectories of the Heavy Ball dynamical system, with constant damping coefficient, in the framework of convex and non-convex smooth optimization. By using the Polyak\u2013\u0141ojasiewicz condition, we derive new linear convergence rates for the associated trajectory, in terms of objective function values, without assuming uniqueness of the minimizer.\n<\/jats:p>","DOI":"10.1007\/s10898-022-01164-w","type":"journal-article","created":{"date-parts":[[2022,5,6]],"date-time":"2022-05-06T04:02:35Z","timestamp":1651809755000},"page":"563-589","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Convergence rates for the heavy-ball continuous dynamics for non-convex optimization, under Polyak\u2013\u0141ojasiewicz condition"],"prefix":"10.1007","volume":"84","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":"Nicol\u00f2","family":"Ginatta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Silvia","family":"Villa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,5,6]]},"reference":[{"key":"1164_CR1","unstructured":"Allen-Zhu, Z., Li, Y., Song, Z.: A convergence theory for deep learning via over-parameterization. In: Chaudhuri, K., Salakhutdinov R. (eds.) Proceedings of the 36th International Conference on Machine Learning, Proceedings of Machine Learning Research, vol.\u00a097, pp. 242\u2013252. PMLR (2019). http:\/\/proceedings.mlr.press\/v97\/allen-zhu19a.html"},{"issue":"4","key":"1164_CR2","doi-asserted-by":"publisher","first-page":"1102","DOI":"10.1137\/S0363012998335802","volume":"38","author":"F Alvarez","year":"2000","unstructured":"Alvarez, 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":"1164_CR3","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1016\/S0021-7824(01)01253-3","volume":"81","author":"F Alvarez","year":"2002","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(8), 747\u2013779 (2002)","journal-title":"J. Math. Pures Appl."},{"key":"1164_CR4","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":"1164_CR5","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":"1164_CR6","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10107-020-01476-3","volume":"187","author":"V Apidopoulos","year":"2021","unstructured":"Apidopoulos, V., Aujol, J.F., Dossal, C., Rondepierre, A.: Convergence rates of an inertial gradient descent algorithm under growth and flatness conditions. Math. Program. 187(1), 151\u2013193 (2021). https:\/\/doi.org\/10.1007\/s10107-020-01476-3","journal-title":"Math. Program."},{"key":"1164_CR7","unstructured":"Attouch, H., Bot, R.I., Csetnek, E.R.: Fast optimization via inertial dynamics with closed-loop damping. arXiv preprint arXiv:2008.02261 (2020)"},{"issue":"9","key":"1164_CR8","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(9), 5412\u20135458 (2017)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"1164_CR9","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":"1","key":"1164_CR10","first-page":"273","volume":"12","author":"H Attouch","year":"2002","unstructured":"Attouch, H., Cabot, A., Redont, P.: The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv. Math. Sci. Appl. 12(1), 273\u2013306 (2002)","journal-title":"Adv. Math. Sci. Appl."},{"key":"1164_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-020-01591-1","author":"H Attouch","year":"2020","unstructured":"Attouch, H., Chbani, Z., Fadili, J., Riahi, H.: First-order optimization algorithms via inertial systems with hessian driven damping. Math. Program. (2020). https:\/\/doi.org\/10.1007\/s10107-020-01591-1","journal-title":"Math. Program."},{"issue":"1\u20132","key":"1164_CR12","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":"1164_CR13","doi-asserted-by":"crossref","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, Optimisation and Calculus of Variations (2019)","DOI":"10.1051\/cocv\/2017083"},{"issue":"01","key":"1164_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1142\/S0219199700000025","volume":"2","author":"H Attouch","year":"2000","unstructured":"Attouch, H., Goudou, X., Redont, P.: The heavy ball with friction method, i. The continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 2(01), 1\u201334 (2000)","journal-title":"Commun. Contemp. Math."},{"key":"1164_CR15","doi-asserted-by":"crossref","unstructured":"Aujol, J.F., Dossal, C., Rondepierre, A.: Optimal convergence rates for Nesterov acceleration. arXiv preprint arXiv:1805.05719 (2018)","DOI":"10.1137\/18M1186757"},{"key":"1164_CR16","unstructured":"Aujol, J.F., Dossal, C., Rondepierre, A.: Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization (2020). https:\/\/hal.archives-ouvertes.fr\/hal-02545245. Working paper or preprint"},{"key":"1164_CR17","unstructured":"Aujol, J.F., Dossal, C., Rondepierre, A.: Convergence rates of the Heavy-Ball method with Lojasiewicz property. Research report, IMB - Institut de Math\u00e9matiques de Bordeaux; INSA Toulouse; UPS Toulouse (2020). https:\/\/hal.archives-ouvertes.fr\/hal-02928958"},{"key":"1164_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)"},{"issue":"7","key":"1164_CR19","doi-asserted-by":"publisher","first-page":"3115","DOI":"10.1016\/j.jde.2015.04.016","volume":"259","author":"P B\u00e9gout","year":"2015","unstructured":"B\u00e9gout, P., Bolte, J., Jendoubi, M.A.: On damped second-order gradient systems. J. Differ. Equ. 259(7), 3115\u20133143 (2015)","journal-title":"J. Differ. Equ."},{"issue":"3","key":"1164_CR20","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1057\/palgrave.jors.2600425","volume":"48","author":"DP Bertsekas","year":"1997","unstructured":"Bertsekas, D.P.: Nonlinear programming. J. Oper. Res. Soc. 48(3), 334\u2013334 (1997). https:\/\/doi.org\/10.1057\/palgrave.jors.2600425","journal-title":"J. Oper. Res. Soc."},{"issue":"4","key":"1164_CR21","doi-asserted-by":"publisher","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, A., Lewis, A.: The \u0142ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205\u20131223 (2007)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"1164_CR22","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 \u0142ojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319\u20133363 (2010)","journal-title":"Trans. Am. Math. Soc."},{"issue":"2","key":"1164_CR23","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":"3","key":"1164_CR24","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1007\/s00028-018-0441-7","volume":"18","author":"RI Bo\u0163","year":"2018","unstructured":"Bo\u0163, R.I., Csetnek, E.R., L\u00e1szl\u00f3, S.C.: Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems. J. Evol. Equ. 18(3), 1291\u20131318 (2018)","journal-title":"J. Evol. Equ."},{"issue":"11","key":"1164_CR25","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":"1164_CR26","first-page":"33","volume":"17","author":"A Cabot","year":"2009","unstructured":"Cabot, A., Engler, H., Gadat, S.: Second-order differential equations with asymptotically small dissipation and piecewise flat potentials. Electron. J. Differ. Equ. 17, 33\u201338 (2009)","journal-title":"Electron. J. Differ. Equ."},{"issue":"3","key":"1164_CR27","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.S.: Error bounds, quadratic growth, and linear convergence of proximal methods. Math. Oper. Res. 43(3), 919\u2013948 (2018)","journal-title":"Math. Oper. Res."},{"key":"1164_CR28","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":"3","key":"1164_CR29","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1137\/15M1029485","volume":"54","author":"M Ghisi","year":"2016","unstructured":"Ghisi, M., Gobbino, M., Haraux, A.: The remarkable effectiveness of time-dependent damping terms for second order evolution equations. SIAM J. Control Optim. 54(3), 1266\u20131294 (2016)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1164_CR30","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1006\/jdeq.1997.3393","volume":"144","author":"A Haraux","year":"1998","unstructured":"Haraux, A., Jendoubi, M.A.: Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. J. Differ. Equ. 144(2), 313\u2013320 (1998)","journal-title":"J. Differ. Equ."},{"key":"1164_CR31","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/978-3-319-46128-1_50","volume-title":"Mach. Learn. Knowl. Discov. Databases","author":"H Karimi","year":"2016","unstructured":"Karimi, H., Nutini, J., Schmidt, M.: Linear convergence of gradient and proximal-gradient methods under the Polyak\u2013\u0141ojasiewicz condition. In: Frasconi, P., Landwehr, N., Manco, G., Vreeken, J. (eds.) Mach. Learn. Knowl. Discov. Databases, pp. 795\u2013811. Springer, Cham (2016)"},{"issue":"3","key":"1164_CR32","doi-asserted-by":"publisher","first-page":"769","DOI":"10.5802\/aif.1638","volume":"48","author":"K Kurdyka","year":"1998","unstructured":"Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48(3), 769\u2013783 (1998)","journal-title":"Ann. Inst. Fourier"},{"key":"1164_CR33","unstructured":"Liu, C., Zhu, L., Belkin, M.: Toward a theory of optimization for over-parameterized systems of non-linear equations: the lessons of deep learning (2020)"},{"key":"1164_CR34","unstructured":"Lojasiewicz, S.: Une propri\u00e9t\u00e9 topologique des sous-ensembles analytiques r\u00e9els, in \u201cles \u00e9quations aux d\u00e9riv\u00e9es partielles (paris, 1962)\u201d \u00e9ditions du centre national de la recherche scientifique (1963)"},{"issue":"1\u20132","key":"1164_CR35","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s10107-018-1232-1","volume":"175","author":"I Necoara","year":"2019","unstructured":"Necoara, I., Nesterov, Y., Glineur, F.: Linear convergence of first order methods for non-strongly convex optimization. Math. Program. 175(1\u20132), 69\u2013107 (2019)","journal-title":"Math. Program."},{"key":"1164_CR36","unstructured":"Nesterov, Y.: Introductory lectures on convex optimization: a basic course (2013)"},{"issue":"1","key":"1164_CR37","doi-asserted-by":"publisher","first-page":"7456","DOI":"10.1016\/j.ifacol.2017.08.1513","volume":"50","author":"B Polyak","year":"2017","unstructured":"Polyak, B., Shcherbakov, P.: Lyapunov functions: an optimization theory perspective. IFAC Papers OnLine 50(1), 7456\u20137461 (2017)","journal-title":"IFAC Papers OnLine"},{"issue":"4","key":"1164_CR38","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":"1164_CR39","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":"1164_CR40","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":"1164_CR41","unstructured":"Siegel, J.W.: Accelerated first-order methods: differential equations and Lyapunov functions. arXiv preprint arXiv:1903.05671 (2019)"},{"issue":"153","key":"1164_CR42","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."},{"key":"1164_CR43","unstructured":"Yuan, Z., Yan, Y., Jin, R., Yang, T.: Stagewise training accelerates convergence of testing error over sgd. In: Wallach, H., Larochelle, H., Beygelzimer, A., Alch\u00e9-Buc, F.D, Fox, E., Garnett R. (eds.) Advances in Neural Information Processing Systems, vol.\u00a032. Curran Associates, Inc (2019). https:\/\/proceedings.neurips.cc\/paper\/2019\/file\/fcdf25d6e191893e705819b177cddea0-Paper.pdf"},{"issue":"1","key":"1164_CR44","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s10107-018-01360-1","volume":"180","author":"H Zhang","year":"2020","unstructured":"Zhang, H.: New analysis of linear convergence of gradient-type methods via unifying error bound conditions. Math. Program. 180(1), 371\u2013416 (2020)","journal-title":"Math. Program."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-022-01164-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-022-01164-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-022-01164-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,20]],"date-time":"2022-10-20T06:12:20Z","timestamp":1666246340000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-022-01164-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,6]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["1164"],"URL":"https:\/\/doi.org\/10.1007\/s10898-022-01164-w","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,6]]},"assertion":[{"value":"23 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}