[{"date_created":"2024-11-28T08:58:17Z","title":"Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization","year":"2025","external_id":{"arxiv":["2411.18422"]},"language":[{"iso":"eng"}],"keyword":["Pareto optimization","Lyapunov analysis","gradient-like dynamical systems","inertial dynamics","asymptotic vanishing damping","Tikhonov regularization","strong convergence"],"ddc":["510"],"publication":"Journal of Mathematical Analysis and Applications","file":[{"relation":"main_file","content_type":"application/pdf","file_id":"57473","access_level":"open_access","file_name":"Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization.pdf","file_size":4291134,"date_created":"2024-11-28T08:58:00Z","creator":"sonntagk","date_updated":"2024-11-28T08:58:00Z"}],"abstract":[{"lang":"eng","text":"In this paper we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions toward a weak Pareto optimal solution. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. This work extends results from single objective convex optimization into the multiobjective setting."}],"author":[{"full_name":"Bot, Radu Ioan","last_name":"Bot","first_name":"Radu Ioan"},{"full_name":"Sonntag, Konstantin","id":"56399","orcid":"https://orcid.org/0000-0003-3384-3496","last_name":"Sonntag","first_name":"Konstantin"}],"oa":"1","date_updated":"2025-10-16T11:56:36Z","main_file_link":[{"url":"https://arxiv.org/pdf/2411.18422"}],"has_accepted_license":"1","citation":{"ama":"Bot RI, Sonntag K. Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization. <i>Journal of Mathematical Analysis and Applications</i>. Published online 2025.","ieee":"R. I. Bot and K. Sonntag, “Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization,” <i>Journal of Mathematical Analysis and Applications</i>, 2025.","chicago":"Bot, Radu Ioan, and Konstantin Sonntag. “Inertial Dynamics with Vanishing Tikhonov Regularization for Multobjective Optimization.” <i>Journal of Mathematical Analysis and Applications</i>, 2025.","bibtex":"@article{Bot_Sonntag_2025, title={Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization}, journal={Journal of Mathematical Analysis and Applications}, author={Bot, Radu Ioan and Sonntag, Konstantin}, year={2025} }","short":"R.I. Bot, K. Sonntag, Journal of Mathematical Analysis and Applications (2025).","mla":"Bot, Radu Ioan, and Konstantin Sonntag. “Inertial Dynamics with Vanishing Tikhonov Regularization for Multobjective Optimization.” <i>Journal of Mathematical Analysis and Applications</i>, 2025.","apa":"Bot, R. I., &#38; Sonntag, K. (2025). Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization. <i>Journal of Mathematical Analysis and Applications</i>."},"department":[{"_id":"101"},{"_id":"530"},{"_id":"655"}],"user_id":"56399","_id":"57472","file_date_updated":"2024-11-28T08:58:00Z","type":"journal_article","status":"public"},{"type":"journal_article","publication":"SIAM Journal on Optimization","abstract":[{"lang":"eng","text":"We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas previously laid out in [H. Attouch and G. Garrigos, Multiobjective Optimization: An Inertial Dynamical Approach to Pareto Optima, preprint, arXiv:1506.02823, 2015]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order \\(\\mathcal{O}(t^{-2})\\) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization."}],"status":"public","_id":"32447","user_id":"56399","department":[{"_id":"101"},{"_id":"655"}],"article_type":"original","keyword":["multiobjective optimization","Pareto optimization","Lyapunov analysis","gradient-likedynamical systems","inertial dynamics","asymptotic vanishing damping","fast convergence"],"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["1095-7189"]},"issue":"3","year":"2024","citation":{"short":"K. Sonntag, S. Peitz, SIAM Journal on Optimization 34 (2024) 2259–2286.","bibtex":"@article{Sonntag_Peitz_2024, title={Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping}, volume={34}, DOI={<a href=\"https://doi.org/10.1137/23M1588512\">10.1137/23M1588512</a>}, number={3}, journal={SIAM Journal on Optimization}, publisher={Society for Industrial and Applied Mathematics}, author={Sonntag, Konstantin and Peitz, Sebastian}, year={2024}, pages={2259–2286} }","mla":"Sonntag, Konstantin, and Sebastian Peitz. “Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping.” <i>SIAM Journal on Optimization</i>, vol. 34, no. 3, Society for Industrial and Applied Mathematics, 2024, pp. 2259–86, doi:<a href=\"https://doi.org/10.1137/23M1588512\">10.1137/23M1588512</a>.","ama":"Sonntag K, Peitz S. Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping. <i>SIAM Journal on Optimization</i>. 2024;34(3):2259-2286. doi:<a href=\"https://doi.org/10.1137/23M1588512\">10.1137/23M1588512</a>","apa":"Sonntag, K., &#38; Peitz, S. (2024). Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping. <i>SIAM Journal on Optimization</i>, <i>34</i>(3), 2259–2286. <a href=\"https://doi.org/10.1137/23M1588512\">https://doi.org/10.1137/23M1588512</a>","chicago":"Sonntag, Konstantin, and Sebastian Peitz. “Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping.” <i>SIAM Journal on Optimization</i> 34, no. 3 (2024): 2259–86. <a href=\"https://doi.org/10.1137/23M1588512\">https://doi.org/10.1137/23M1588512</a>.","ieee":"K. Sonntag and S. Peitz, “Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping,” <i>SIAM Journal on Optimization</i>, vol. 34, no. 3, pp. 2259–2286, 2024, doi: <a href=\"https://doi.org/10.1137/23M1588512\">10.1137/23M1588512</a>."},"intvolume":"        34","page":"2259 - 2286","publisher":"Society for Industrial and Applied Mathematics","date_updated":"2024-07-02T09:27:39Z","author":[{"first_name":"Konstantin","orcid":"https://orcid.org/0000-0003-3384-3496","last_name":"Sonntag","full_name":"Sonntag, Konstantin","id":"56399"},{"orcid":"0000-0002-3389-793X","last_name":"Peitz","full_name":"Peitz, Sebastian","id":"47427","first_name":"Sebastian"}],"date_created":"2022-07-28T11:53:02Z","volume":34,"title":"Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping","doi":"10.1137/23M1588512"}]