{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T14:19:13Z","timestamp":1780582753131,"version":"3.54.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,9,22]],"date-time":"2024-09-22T00:00:00Z","timestamp":1726963200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,9,22]],"date-time":"2024-09-22T00:00:00Z","timestamp":1726963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP200101197"],"award-info":[{"award-number":["DP200101197"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP160100854"],"award-info":[{"award-number":["DP160100854"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Royal Melbourne Institute of Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka\u2013\u0141ojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.<\/jats:p>","DOI":"10.1007\/s10957-024-02536-w","type":"journal-article","created":{"date-parts":[[2024,9,22]],"date-time":"2024-09-22T10:01:39Z","timestamp":1726999299000},"page":"844-879","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6426-5727","authenticated-orcid":false,"given":"Lien T.","family":"Nguyen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew","family":"Eberhard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinghuo","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexander Y.","family":"Kruger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chaojie","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,9,22]]},"reference":[{"issue":"2","key":"2536_CR1","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/s10957-013-0414-5","volume":"161","author":"B Abbas","year":"2014","unstructured":"Abbas, B., Attouch, H., Svaiter, B.F.: Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces. J. Optim. Theory Appl. 161(2), 331\u2013360 (2014)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2536_CR2","first-page":"281","volume":"28","author":"S Adly","year":"2021","unstructured":"Adly, S., Attouch, H.: Finite time stabilization of continuous inertial dynamics combining dry friction with Hessian-driven damping. J. Conv. Anal. 28(2), 281\u2013310 (2021)","journal-title":"J. Conv. Anal."},{"key":"2536_CR3","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1007\/0-387-29195-4_24","volume-title":"Nonsmooth Mechanics and Analysis","author":"S Adly","year":"2006","unstructured":"Adly, S., Attouch, H., Cabot, A.: Finite time stabilization of nonlinear oscillators subject to dry friction. In: Alart, P., Maisonneuve, O., Rockafellar, R.T. (eds.) Nonsmooth Mechanics and Analysis, vol. 12, pp. 289\u2013304. Springer, New York (2006)"},{"key":"2536_CR4","doi-asserted-by":"publisher","DOI":"10.1142\/1988","volume-title":"Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations","author":"RP Agarwal","year":"1993","unstructured":"Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, River Edge (1993)"},{"key":"2536_CR5","doi-asserted-by":"publisher","first-page":"779","DOI":"10.2307\/1911819","volume":"29","author":"KJ Arrow","year":"1961","unstructured":"Arrow, K.J., Enthoven, A.C.: Quasiconcave programming. Econometrica 29, 779\u2013800 (1961)","journal-title":"Econometrica"},{"key":"2536_CR6","doi-asserted-by":"crossref","unstructured":"Attouch, H., Az\u00e9, D.: Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry\u2013Lions method. Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire., 10(3), 289\u2013312 (1993)","DOI":"10.1016\/s0294-1449(16)30214-1"},{"issue":"1\u20132","key":"2536_CR7","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116(1\u20132), 5\u201316 (2009)","journal-title":"Math. Program."},{"issue":"2","key":"2536_CR8","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s10898-021-01057-4","volume":"82","author":"S Bartz","year":"2022","unstructured":"Bartz, S., Dao, M.N., Phan, H.M.: Conical averagedness and convergence analysis of fixed point algorithms. J. Glob. Optim. 82(2), 351\u2013373 (2022)","journal-title":"J. Glob. Optim."},{"key":"2536_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-48311-5","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Cham (2017)","edition":"2"},{"issue":"1","key":"2536_CR10","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":"2536_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jmaa.2004.06.003","volume":"303","author":"F Bernard","year":"2005","unstructured":"Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303, 1\u201314 (2005)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"2536_CR12","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1137\/S0363012997321358","volume":"38","author":"SP Bhat","year":"2000","unstructured":"Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751\u2013766 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"2536_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03718-8","volume-title":"Real Algebraic Geometry","author":"J Bochnak","year":"1998","unstructured":"Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer-Verlag, Berlin (1998)"},{"issue":"4","key":"2536_CR14","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 \u0141ojasiewicz 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":"3","key":"2536_CR15","doi-asserted-by":"publisher","first-page":"1423","DOI":"10.1137\/15M1012657","volume":"54","author":"R Bo\u0163","year":"2016","unstructured":"Bo\u0163, R., Csetnek, E.R.: Second order forward-backward dynamical systems for monotone inclusion problems. SIAM J. Optim. 54(3), 1423\u20131443 (2016)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"2536_CR16","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1080\/00036811.2018.1495330","volume":"99","author":"R Bo\u0163","year":"2020","unstructured":"Bo\u0163, R., Csetnek, E.R., L\u00e1szl\u00f3, S.: A second-order dynamical approach with variable damping to nonconvex smooth minimization. Appl Anal. 99(3), 361\u2013378 (2020)","journal-title":"Appl Anal."},{"key":"2536_CR17","doi-asserted-by":"crossref","unstructured":"Chen, F., Ren, W.: Convex optimization via finite-time projected gradient flows. In: IEEE Conference on Decision and Control, pp. 4072\u20134077. Miami (2018)","DOI":"10.1109\/CDC.2018.8619081"},{"issue":"11","key":"2536_CR18","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1016\/j.automatica.2006.06.015","volume":"42","author":"J Cortes","year":"2006","unstructured":"Cortes, J.: Finite-time convergent gradient flows with applications to network consensus. Automatica 42(11), 1993\u20132000 (2006)","journal-title":"Automatica"},{"issue":"4","key":"2536_CR19","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s11228-020-00548-y","volume":"28","author":"ER Csetnek","year":"2020","unstructured":"Csetnek, E.R.: Continuous dynamics related to monotone inclusions and non-smooth optimization problems. Set-Valued Var. Anal. 28(4), 611\u2013642 (2020)","journal-title":"Set-Valued Var. Anal."},{"issue":"5","key":"2536_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10444-021-09891-6","volume":"47","author":"ER Csetnek","year":"2021","unstructured":"Csetnek, E.R., Eberhard, A., Tam, M.K.: Convergence rates for boundedly regular systems. Adv. Comput. Math. 47(5), 1\u201318 (2021)","journal-title":"Adv. Comput. Math."},{"issue":"8","key":"2536_CR21","doi-asserted-by":"publisher","first-page":"5029","DOI":"10.1109\/TAC.2022.3214795","volume":"68","author":"K Garg","year":"2023","unstructured":"Garg, K., Baranwal, M., Gupta, R., Benosman, M.: Fixed-time stable proximal dynamical system for solving MVIPs. IEEE Trans. Automat. Control. 68(8), 5029\u20135036 (2023)","journal-title":"IEEE Trans. Automat. Control."},{"issue":"5","key":"2536_CR22","doi-asserted-by":"publisher","first-page":"2002","DOI":"10.1109\/TAC.2020.3001436","volume":"66","author":"K Garg","year":"2020","unstructured":"Garg, K., Panagou, D.: Fixed-time stable gradient flows: applications to continuous-time optimization. IEEE Trans. Automat. Control. 66(5), 2002\u20132015 (2020)","journal-title":"IEEE Trans. Automat. Control."},{"key":"2536_CR23","doi-asserted-by":"crossref","unstructured":"Haddad, W.M., Nersesov, S.G., Du, L.: Finite-time stability for time-varying nonlinear dynamical systems. In: Proceedings of the American Control Conference pp. 4135\u20134139. Seattle (2008)","DOI":"10.1109\/ACC.2008.4587141"},{"issue":"4","key":"2536_CR24","doi-asserted-by":"publisher","first-page":"760","DOI":"10.1137\/0324047","volume":"24","author":"V Haimo","year":"1986","unstructured":"Haimo, V.: Finite time controllers. SIAM J. Control Optim. 24(4), 760\u2013770 (1986)","journal-title":"SIAM J. Control Optim."},{"key":"2536_CR25","volume-title":"Ordinary Differential Equations","author":"JK Hale","year":"1980","unstructured":"Hale, J.K.: Ordinary Differential Equations, 2nd edn. Robert E. Krieger, Huntington (1980)","edition":"2"},{"key":"2536_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8134-0","volume-title":"A Primer of Real Analytic Functions","author":"SG Krantz","year":"2002","unstructured":"Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, 2nd edn. Birkh\u00e4user, Boston (2002)","edition":"2"},{"issue":"44","key":"2536_CR27","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1007\/s11228-019-0504-0","volume":"27","author":"AY Kruger","year":"2019","unstructured":"Kruger, A.Y., L\u00f3pez, M.A., Yang, X.Q., Zhu, J.X.: H\u00f6lder error bounds and H\u00f6lder calmness with applications to convex semi-infinite optimization. Set Valued Var. Anal. 27(44), 995\u20131023 (2019)","journal-title":"Set Valued Var. Anal."},{"key":"2536_CR28","volume-title":"Nonlinear Differential Equations in Abstract Spaces","author":"V Lakshmikantham","year":"1981","unstructured":"Lakshmikantham, V., Leela, S.: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981)"},{"issue":"5","key":"2536_CR29","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1007\/s10208-017-9366-8","volume":"18","author":"G Li","year":"2018","unstructured":"Li, G., Pong, T.K.: Calculus of the exponent of Kurdyka-\u0141ojasiewicz inequality and its applications to linear convergence of first-order methods. Found. Comput. Math. 18(5), 1199\u20131232 (2018)","journal-title":"Found. Comput. Math."},{"key":"2536_CR30","doi-asserted-by":"crossref","unstructured":"Li, C., Yu, X., Zhou, X., Ren, W.: A fixed time distributed optimization: a sliding mode perspective. In: 43rd Annual Conference of the IEEE Industrial Electronics Society, pp. 8201\u20138207. Beijing (2017)","DOI":"10.1109\/IECON.2017.8217439"},{"key":"2536_CR31","unstructured":"\u0141ojasiewicz, S.: Une propri\u00e9t\u00e9 topologique des sous-ensembles analytiques r\u00e9els, Les \u00c9quations aux D\u00e9riv\u00e9es Partielles, pp. 87\u201389. \u00c9ditions du Centre National de la Recherche Scientifique, Paris (1963)"},{"key":"2536_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-91578-4","volume-title":"Lectures on Convex Optimization","author":"YE Nesterov","year":"2018","unstructured":"Nesterov, Y.E.: Lectures on Convex Optimization. Springer, Cham (2018)"},{"issue":"3","key":"2536_CR33","doi-asserted-by":"publisher","first-page":"2557","DOI":"10.1137\/22M1532354","volume":"34","author":"LT Nguyen","year":"2024","unstructured":"Nguyen, L.T., Eberhard, A., Yu, X., Li, C.: Fast gradient algorithm with dry-like friction and nonmonotone line search for nonconvex optimization problems. SIAM J. Optim. 34(3), 2557\u20132587 (2024)","journal-title":"SIAM J. Optim."},{"issue":"7","key":"2536_CR34","first-page":"4383","volume":"68","author":"LT Nguyen","year":"2023","unstructured":"Nguyen, L.T., Yu, X., Eberhard, A., Li, C.: Fixed-time gradient dynamics with time-varying coefficients for continuous-time optimization. IEEE Trans. Automat. Control. 68(7), 4383\u20134390 (2023)","journal-title":"IEEE Trans. Automat. Control."},{"key":"2536_CR35","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1090\/S0002-9904-1967-11761-0","volume":"73","author":"Z Opial","year":"1967","unstructured":"Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591\u2013597 (1967)","journal-title":"Bull. Am. Math. Soc."},{"issue":"8","key":"2536_CR36","doi-asserted-by":"publisher","first-page":"2106","DOI":"10.1109\/TAC.2011.2179869","volume":"57","author":"A Polyakov","year":"2012","unstructured":"Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Automat. Control. 57(8), 2106\u20132110 (2012)","journal-title":"IEEE Trans. Automat. Control."},{"key":"2536_CR37","unstructured":"Romero, O., Benosman, M.: Finite-time convergence in continuous-time optimization. In: Hal Daum\u00e9 III, Aarti Singh (eds.) Proceedings of the 37th International Conference on Machine Learning, PMLR, vol. 119, pp. 8200\u20138209 (2020)"},{"key":"2536_CR38","unstructured":"Su, W., Boyd, S., Cand\u00e8s, E.: A differential equation for modeling Nesterov\u2019s accelerated gradient method: theory and insights. Adv. Neural Inf. Process. Syst., 2510\u20132518 (2014)"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02536-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02536-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02536-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T18:05:57Z","timestamp":1730484357000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02536-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,22]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2536"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02536-w","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,22]]},"assertion":[{"value":"27 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 September 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}}]}}