{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T19:22:59Z","timestamp":1777490579871,"version":"3.51.4"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["project W 1260"],"award-info":[{"award-number":["project W 1260"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Austrian Science Fund"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution\u2014the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.<\/jats:p>","DOI":"10.1007\/s10589-023-00536-6","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T10:02:19Z","timestamp":1698228139000},"page":"531-569","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique"],"prefix":"10.1007","volume":"87","author":[{"given":"Mikhail A.","family":"Karapetyants","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,10,25]]},"reference":[{"key":"536_CR1","doi-asserted-by":"publisher","first-page":"7138","DOI":"10.1016\/j.jde.2018.02.017","volume":"264","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Cabot, A.: Convergence of damped inertial dynamics governed by regularized maximally monotone operators. J. Differ. Equ. 264, 7138\u20137182 (2018)","journal-title":"J. Differ. Equ."},{"key":"536_CR2","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/j.jde.2021.12.005","volume":"311","author":"H Attouch","year":"2022","unstructured":"Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Damped inertial dynamics with vanishing Tikhonov regularization: strong asymptotic convergence towards the minimum norm solution. J. Differ. Equ. 311, 29\u201358 (2022)","journal-title":"J. Differ. Equ."},{"key":"536_CR3","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s00245-023-09997-x","volume":"88","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessian. Appl. Math. Optim. 88, 29 (2023)","journal-title":"Appl. Math. Optim."},{"key":"536_CR4","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":"536_CR5","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/s11228-020-00564-y","volume":"29","author":"H Attouch","year":"2021","unstructured":"Attouch, H., L\u00e1szl\u00f3, S.C.: Continuous Newton-like inertial dynamics for monotone inclusions. Set-valued Var. Anal. 29, 555\u2013581 (2021)","journal-title":"Set-valued Var. Anal."},{"key":"536_CR6","doi-asserted-by":"crossref","unstructured":"Attouch, H., L\u00e1szl\u00f3, S.C.: Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution. arXiv:2104.11987 (2021)","DOI":"10.1016\/j.jde.2021.12.005"},{"key":"536_CR7","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s10107-018-1252-x","volume":"174","author":"H Attouch","year":"2019","unstructured":"Attouch, H., Peypouquet, J.: Convergence of the inertial dynamics and proximal algorithms governed by maximally monotone operators. Math. Program. 174, 391\u2013432 (2019)","journal-title":"Math. Program."},{"issue":"10","key":"536_CR8","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 damping. J. Differ. Equ. 261(10), 5734\u20135783 (2016)","journal-title":"J. Differ. Equ."},{"key":"536_CR9","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics","author":"HH Bauschke","year":"2016","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, Berlin (2016)"},{"key":"536_CR10","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10107-020-01528-8","volume":"189","author":"RI Bo\u0163","year":"2021","unstructured":"Bo\u0163, 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."},{"issue":"2","key":"536_CR11","doi-asserted-by":"publisher","first-page":"127689","DOI":"10.1016\/j.jmaa.2023.127689","volume":"530","author":"RI Bo\u0163","year":"2023","unstructured":"Bo\u0163, R.I., Csetnek, E.R., L\u00e1szl\u00f3, S.C.: On the strong convergence of continuous Newton-like inertial dynamics with Tikhonov regularization for monotone inclusions. J. Math. Anal. Appl. 530(2), 127689 (2023)","journal-title":"J. Math. Anal. Appl."},{"key":"536_CR12","doi-asserted-by":"crossref","unstructured":"Bo\u0163, R.I., Karapetyants, M.A.: A fast continuous time approach with time scaling for nonsmooth convex optimization. In: Advances in Continuous and Discrete Models: Theory and Applications, vol. 73 (2022)","DOI":"10.1186\/s13662-022-03744-2"},{"key":"536_CR13","doi-asserted-by":"crossref","unstructured":"Csetnek, R.E., Karapetyants, M.A.: A fast continuous time approach for non-smooth convex optimization with time scaling and Tikhonov regularization, preprint (2022)","DOI":"10.1186\/s13662-022-03744-2"},{"key":"536_CR14","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.jde.2023.03.014","volume":"362","author":"SC L\u00e1szl\u00f3","year":"2022","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 (2022)","journal-title":"J. Differ. Equ."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00536-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-023-00536-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00536-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T13:07:28Z","timestamp":1708088848000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-023-00536-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,25]]},"references-count":14,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["536"],"URL":"https:\/\/doi.org\/10.1007\/s10589-023-00536-6","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,25]]},"assertion":[{"value":"12 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}