A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique
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 M...
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| Veröffentlicht in: | Computational optimization and applications Jg. 87; H. 2; S. 531 - 569 |
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| Sprache: | Englisch |
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| Abstract | 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—the 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. |
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| AbstractList | 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-the 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.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-the 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. 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—the 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. |
| Author | Karapetyants, Mikhail A. |
| Author_xml | – sequence: 1 givenname: Mikhail A. surname: Karapetyants fullname: Karapetyants, Mikhail A. email: mikhail.karapetyants@univie.ac.at organization: Faculty of Mathematics, University of Vienna |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/38357400$$D View this record in MEDLINE/PubMed |
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| CitedBy_id | crossref_primary_10_1080_10556788_2025_2517172 crossref_primary_10_1007_s00245_024_10163_0 crossref_primary_10_1007_s10957_025_02766_6 crossref_primary_10_1016_j_cam_2024_116394 crossref_primary_10_1080_02331934_2025_2534125 crossref_primary_10_1007_s11075_025_02010_2 crossref_primary_10_1080_02331934_2025_2520445 |
| Cites_doi | 10.1016/j.jde.2018.02.017 10.1016/j.jde.2016.08.020 10.1016/j.jmaa.2023.127689 10.1016/j.jmaa.2016.12.017 10.1007/s10107-018-1252-x 10.1007/s10107-020-01528-8 10.1016/j.jde.2021.12.005 10.1007/s00245-023-09997-x 10.1016/j.jde.2023.03.014 10.1007/s11228-020-00564-y 10.1186/s13662-022-03744-2 |
| ContentType | Journal Article |
| Copyright | The Author(s) 2023 The Author(s) 2023. The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Keywords | Strong convergence 65K05 Nonsmooth convex optimization Moreau envelope Proximal operator Hessian-driven damping Tikhonov regularization 37N40 90C25 Damped inertial dynamics 65K10 46N10 49M99 |
| Language | English |
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| SubjectTerms | Convergence Convex analysis Convex and Discrete Geometry Convexity Damping Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Regularization Statistics |
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| Title | A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique |
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