[{"type":"journal_article","ddc":["510"],"publication":"Journal of Mathematical Analysis and Applications","abstract":[{"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.","lang":"eng"}],"has_accepted_license":"1","title":"Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization","file":[{"content_type":"application/pdf","file_id":"57473","access_level":"open_access","date_created":"2024-11-28T08:58:00Z","creator":"sonntagk","file_name":"Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization.pdf","file_size":4291134,"date_updated":"2024-11-28T08:58:00Z","relation":"main_file"}],"external_id":{"arxiv":["2411.18422"]},"user_id":"56399","oa":"1","main_file_link":[{"url":"https://arxiv.org/pdf/2411.18422"}],"keyword":["Pareto optimization","Lyapunov analysis","gradient-like dynamical systems","inertial dynamics","asymptotic vanishing damping","Tikhonov regularization","strong convergence"],"year":"2025","language":[{"iso":"eng"}],"status":"public","date_created":"2024-11-28T08:58:17Z","date_updated":"2025-10-16T11:56:36Z","_id":"57472","file_date_updated":"2024-11-28T08:58:00Z","author":[{"first_name":"Radu Ioan","full_name":"Bot, Radu Ioan","last_name":"Bot"},{"orcid":"https://orcid.org/0000-0003-3384-3496","first_name":"Konstantin","full_name":"Sonntag, Konstantin","id":"56399","last_name":"Sonntag"}],"department":[{"_id":"101"},{"_id":"530"},{"_id":"655"}],"citation":{"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.","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.","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>.","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.","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.","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} }"}},{"author":[{"full_name":"Sonntag, Konstantin","first_name":"Konstantin","last_name":"Sonntag","id":"56399","orcid":"https://orcid.org/0000-0003-3384-3496"}],"title":"First-order methods and gradient dynamical systems for multiobjective optimization","abstract":[{"lang":"eng","text":"Diese Dissertation enthält Beiträge zum Bereich der Mehrzieloptimierung mit einem Fokus auf unbeschränkten Problemen, die auf einem allgemeinen Hilbertraum definiert sind. Für Mehrzieloptimierungsprobleme mit lokal Lipschitz-stetigen Zielfunktionen definieren wir ein multikriterielles Subdifferential, das wir erstmals im Kontext allgemeiner Hilberträume analysieren. Aufbauend auf diesen theoretischen Untersuchungen präsentieren wir ein Abstiegsverfahren, bei welchem in jeder Iteration eine Abstiegsrichtung mittels einer numerischen Approximation des multikriteriellen Subdifferentials bestimmt wird. Im Kontext konvexer, stetig differenzierbarer Zielfunktionen mit Lipschitz-stetigen Gradienten, führen wir eine Familie von dynamischen Gradientensystemen mit Trägheitsterm ein, die bekannte kontinuierliche Systeme aus der skalaren Optimierung verallgemeinern. Wir stellen drei neue Systeme vor: eines mit konstanter Dämpfung, eines mit asymptotisch abnehmender Dämpfung und eines, das zusätzlich eine zeitabhängige Tikhonov-Regularisierung beinhaltet. Aufbauend auf den Untersuchungen der neuen dynamischen Gradientensysteme, entwickeln wir ein beschleunigtes Gradientenverfahren zur Mehrzieloptimierung, das auf einer Diskretisierung des multikriteriellen Gradientensystems mit asymptotisch abnehmender Dämpfung beruht. Das hergeleitete Verfahren bewahrt die günstigen Konvergenzeigenschaften des kontinuierlichen Systems und erreicht eine schnellere Konvergenz als klassische Verfahren."},{"lang":"eng","text":"This dissertation contributes to the field of multiobjective optimization, with a focus on unconstrained problems formulated in a general Hilbert space. For multiobjective optimization problems with locally Lipschitz continuous objective functions, we define a multiobjective subdifferential, which we analyze for the first time in the context of general Hilbert spaces. Building on these theoretical investigations, we present a descent method in which, at each iteration, a descent direction is determined via a numerical approximation of the multiobjective subdifferential. In the setting of convex, continuously differentiable objective functions with Lipschitz continuous gradients, we introduce a family of inertial gradient dynamical systems that generalize well-known continuous-time systems from scalar optimization. We present three novel systems: one with constant damping, one with asymptotic vanishing damping, and one combining vanishing damping with time-dependent Tikhonov regularization. Building on the investigation of the novel gradient dynamical systems, we develop an accelerated gradient method for multiobjective optimization via discretization of the multiobjective gradient system with asymptotic vanishing damping. The proposed method retains the favorable convergence properties of the continuous system while achieving faster convergence than standard approaches, such as classical methods."}],"doi":"10.17619/UNIPB/1-2457","has_accepted_license":"1","supervisor":[{"full_name":"Dellnitz, Michael","first_name":"Michael","last_name":"Dellnitz"},{"id":"16494","last_name":"Ober-Blöbaum","first_name":"Sina","full_name":"Ober-Blöbaum, Sina"}],"main_file_link":[{"open_access":"1","url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/8141881"}],"user_id":"56399","oa":"1","citation":{"short":"K. Sonntag, First-Order Methods and Gradient Dynamical Systems for Multiobjective Optimization, Paderborn University, 2025.","mla":"Sonntag, Konstantin. <i>First-Order Methods and Gradient Dynamical Systems for Multiobjective Optimization</i>. Paderborn University, 2025, doi:<a href=\"https://doi.org/10.17619/UNIPB/1-2457\">10.17619/UNIPB/1-2457</a>.","bibtex":"@book{Sonntag_2025, title={First-order methods and gradient dynamical systems for multiobjective optimization}, DOI={<a href=\"https://doi.org/10.17619/UNIPB/1-2457\">10.17619/UNIPB/1-2457</a>}, publisher={Paderborn University}, author={Sonntag, Konstantin}, year={2025} }","chicago":"Sonntag, Konstantin. <i>First-Order Methods and Gradient Dynamical Systems for Multiobjective Optimization</i>. Paderborn University, 2025. <a href=\"https://doi.org/10.17619/UNIPB/1-2457\">https://doi.org/10.17619/UNIPB/1-2457</a>.","ieee":"K. Sonntag, <i>First-order methods and gradient dynamical systems for multiobjective optimization</i>. Paderborn University, 2025.","ama":"Sonntag K. <i>First-Order Methods and Gradient Dynamical Systems for Multiobjective Optimization</i>. Paderborn University; 2025. doi:<a href=\"https://doi.org/10.17619/UNIPB/1-2457\">10.17619/UNIPB/1-2457</a>","apa":"Sonntag, K. (2025). <i>First-order methods and gradient dynamical systems for multiobjective optimization</i>. Paderborn University. <a href=\"https://doi.org/10.17619/UNIPB/1-2457\">https://doi.org/10.17619/UNIPB/1-2457</a>"},"department":[{"_id":"101"},{"_id":"530"}],"publisher":"Paderborn University","date_created":"2025-12-03T06:55:01Z","ddc":["510"],"status":"public","language":[{"iso":"eng"}],"type":"dissertation","year":"2025","_id":"62750","date_updated":"2025-12-03T07:04:36Z"},{"year":"2022","language":[{"iso":"eng"}],"status":"public","date_created":"2020-12-15T07:46:36Z","publisher":"IEEE","date_updated":"2022-10-21T12:27:16Z","_id":"20731","file_date_updated":"2021-09-25T11:59:15Z","intvolume":"        44","article_type":"original","author":[{"last_name":"Bieker","id":"32829","first_name":"Katharina","full_name":"Bieker, Katharina"},{"full_name":"Gebken, Bennet","first_name":"Bennet","last_name":"Gebken","id":"32643"},{"orcid":"0000-0002-3389-793X","id":"47427","last_name":"Peitz","full_name":"Peitz, Sebastian","first_name":"Sebastian"}],"department":[{"_id":"101"},{"_id":"530"},{"_id":"655"}],"citation":{"bibtex":"@article{Bieker_Gebken_Peitz_2022, title={On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}, volume={44}, DOI={<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>}, number={11}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={IEEE}, author={Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian}, year={2022}, pages={7797–7808} }","mla":"Bieker, Katharina, et al. “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation.” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, vol. 44, no. 11, IEEE, 2022, pp. 7797–808, doi:<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>.","short":"K. Bieker, B. Gebken, S. Peitz, IEEE Transactions on Pattern Analysis and Machine Intelligence 44 (2022) 7797–7808.","ama":"Bieker K, Gebken B, Peitz S. On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation. <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>. 2022;44(11):7797-7808. doi:<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>","apa":"Bieker, K., Gebken, B., &#38; Peitz, S. (2022). On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation. <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, <i>44</i>(11), 7797–7808. <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">https://doi.org/10.1109/TPAMI.2021.3114962</a>","ieee":"K. Bieker, B. Gebken, and S. Peitz, “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation,” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, vol. 44, no. 11, pp. 7797–7808, 2022, doi: <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>.","chicago":"Bieker, Katharina, Bennet Gebken, and Sebastian Peitz. “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation.” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i> 44, no. 11 (2022): 7797–7808. <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">https://doi.org/10.1109/TPAMI.2021.3114962</a>."},"publication_status":"epub_ahead","type":"journal_article","publication":"IEEE Transactions on Pattern Analysis and Machine Intelligence","ddc":["510"],"issue":"11","page":"7797-7808","volume":44,"doi":"10.1109/TPAMI.2021.3114962","has_accepted_license":"1","abstract":[{"text":"We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.","lang":"eng"}],"title":"On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation","file":[{"relation":"main_file","date_updated":"2021-09-25T11:59:15Z","success":1,"creator":"speitz","file_name":"On_the_Treatment_of_Optimization_Problems_with_L1_Penalty_Terms_via_Multiobjective_Continuation.pdf","file_size":7990831,"content_type":"application/pdf","file_id":"25040","access_level":"closed","date_created":"2021-09-25T11:59:15Z"}],"oa":"1","user_id":"47427","main_file_link":[{"open_access":"1","url":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9547772"}]}]