[{"language":[{"iso":"eng"}],"keyword":["Applied Mathematics","Computational Mathematics","Control and Optimization"],"department":[{"_id":"101"}],"user_id":"32643","_id":"51208","status":"public","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.</jats:p>"}],"publication":"Computational Optimization and Applications","type":"journal_article","doi":"10.1007/s10589-024-00552-0","title":"A note on the convergence of deterministic gradient sampling in nonsmooth optimization","author":[{"last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643","first_name":"Bennet"}],"date_created":"2024-02-07T07:23:23Z","date_updated":"2024-02-08T08:05:54Z","publisher":"Springer Science and Business Media LLC","citation":{"bibtex":"@article{Gebken_2024, title={A note on the convergence of deterministic gradient sampling in nonsmooth optimization}, DOI={<a href=\"https://doi.org/10.1007/s10589-024-00552-0\">10.1007/s10589-024-00552-0</a>}, journal={Computational Optimization and Applications}, publisher={Springer Science and Business Media LLC}, author={Gebken, Bennet}, year={2024} }","short":"B. Gebken, Computational Optimization and Applications (2024).","mla":"Gebken, Bennet. “A Note on the Convergence of Deterministic Gradient Sampling in Nonsmooth Optimization.” <i>Computational Optimization and Applications</i>, Springer Science and Business Media LLC, 2024, doi:<a href=\"https://doi.org/10.1007/s10589-024-00552-0\">10.1007/s10589-024-00552-0</a>.","apa":"Gebken, B. (2024). A note on the convergence of deterministic gradient sampling in nonsmooth optimization. <i>Computational Optimization and Applications</i>. <a href=\"https://doi.org/10.1007/s10589-024-00552-0\">https://doi.org/10.1007/s10589-024-00552-0</a>","ieee":"B. Gebken, “A note on the convergence of deterministic gradient sampling in nonsmooth optimization,” <i>Computational Optimization and Applications</i>, 2024, doi: <a href=\"https://doi.org/10.1007/s10589-024-00552-0\">10.1007/s10589-024-00552-0</a>.","chicago":"Gebken, Bennet. “A Note on the Convergence of Deterministic Gradient Sampling in Nonsmooth Optimization.” <i>Computational Optimization and Applications</i>, 2024. <a href=\"https://doi.org/10.1007/s10589-024-00552-0\">https://doi.org/10.1007/s10589-024-00552-0</a>.","ama":"Gebken B. A note on the convergence of deterministic gradient sampling in nonsmooth optimization. <i>Computational Optimization and Applications</i>. Published online 2024. doi:<a href=\"https://doi.org/10.1007/s10589-024-00552-0\">10.1007/s10589-024-00552-0</a>"},"year":"2024","publication_identifier":{"issn":["0926-6003","1573-2894"]},"publication_status":"published"},{"language":[{"iso":"eng"}],"user_id":"56399","department":[{"_id":"101"},{"_id":"655"}],"_id":"51334","external_id":{"arxiv":["\t2402.06376"]},"status":"public","abstract":[{"lang":"eng","text":"The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem."}],"type":"preprint","publication":"arXiv:2402.06376","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2402.06376"}],"title":"A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces","date_created":"2024-02-13T09:35:26Z","author":[{"last_name":"Sonntag","orcid":"https://orcid.org/0000-0003-3384-3496","full_name":"Sonntag, Konstantin","id":"56399","first_name":"Konstantin"},{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"first_name":"Georg","full_name":"Müller, Georg","last_name":"Müller"},{"first_name":"Sebastian","id":"47427","full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz"},{"last_name":"Volkwein","full_name":"Volkwein, Stefan","first_name":"Stefan"}],"oa":"1","date_updated":"2024-02-21T10:21:03Z","citation":{"ama":"Sonntag K, Gebken B, Müller G, Peitz S, Volkwein S. A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces. <i>arXiv:240206376</i>. Published online 2024.","ieee":"K. Sonntag, B. Gebken, G. Müller, S. Peitz, and S. Volkwein, “A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces,” <i>arXiv:2402.06376</i>. 2024.","chicago":"Sonntag, Konstantin, Bennet Gebken, Georg Müller, Sebastian Peitz, and Stefan Volkwein. “A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces.” <i>ArXiv:2402.06376</i>, 2024.","short":"K. Sonntag, B. Gebken, G. Müller, S. Peitz, S. Volkwein, ArXiv:2402.06376 (2024).","mla":"Sonntag, Konstantin, et al. “A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces.” <i>ArXiv:2402.06376</i>, 2024.","bibtex":"@article{Sonntag_Gebken_Müller_Peitz_Volkwein_2024, title={A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces}, journal={arXiv:2402.06376}, author={Sonntag, Konstantin and Gebken, Bennet and Müller, Georg and Peitz, Sebastian and Volkwein, Stefan}, year={2024} }","apa":"Sonntag, K., Gebken, B., Müller, G., Peitz, S., &#38; Volkwein, S. (2024). A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces. In <i>arXiv:2402.06376</i>."},"year":"2024","has_accepted_license":"1"},{"abstract":[{"lang":"eng","text":"Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 \"Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction\" of the DFG Priority Programm 1962 \"Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization\"."}],"status":"public","type":"preprint","publication":"arXiv:2308.01113","language":[{"iso":"eng"}],"_id":"46578","external_id":{"arxiv":["2308.01113"]},"user_id":"47427","department":[{"_id":"655"},{"_id":"101"}],"year":"2023","citation":{"chicago":"Bernreuther, Marco, Michael Dellnitz, Bennet Gebken, Georg Müller, Sebastian Peitz, Konstantin Sonntag, and Stefan Volkwein. “Multiobjective Optimization of Non-Smooth PDE-Constrained Problems.” <i>ArXiv:2308.01113</i>, 2023.","ieee":"M. Bernreuther <i>et al.</i>, “Multiobjective Optimization of Non-Smooth PDE-Constrained Problems,” <i>arXiv:2308.01113</i>. 2023.","ama":"Bernreuther M, Dellnitz M, Gebken B, et al. Multiobjective Optimization of Non-Smooth PDE-Constrained Problems. <i>arXiv:230801113</i>. Published online 2023.","apa":"Bernreuther, M., Dellnitz, M., Gebken, B., Müller, G., Peitz, S., Sonntag, K., &#38; Volkwein, S. (2023). Multiobjective Optimization of Non-Smooth PDE-Constrained Problems. In <i>arXiv:2308.01113</i>.","short":"M. Bernreuther, M. Dellnitz, B. Gebken, G. Müller, S. Peitz, K. Sonntag, S. Volkwein, ArXiv:2308.01113 (2023).","mla":"Bernreuther, Marco, et al. “Multiobjective Optimization of Non-Smooth PDE-Constrained Problems.” <i>ArXiv:2308.01113</i>, 2023.","bibtex":"@article{Bernreuther_Dellnitz_Gebken_Müller_Peitz_Sonntag_Volkwein_2023, title={Multiobjective Optimization of Non-Smooth PDE-Constrained Problems}, journal={arXiv:2308.01113}, author={Bernreuther, Marco and Dellnitz, Michael and Gebken, Bennet and Müller, Georg and Peitz, Sebastian and Sonntag, Konstantin and Volkwein, Stefan}, year={2023} }"},"title":"Multiobjective Optimization of Non-Smooth PDE-Constrained Problems","main_file_link":[{"url":"https://arxiv.org/pdf/2308.01113","open_access":"1"}],"oa":"1","date_updated":"2024-02-21T12:22:20Z","author":[{"last_name":"Bernreuther","full_name":"Bernreuther, Marco","first_name":"Marco"},{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"first_name":"Bennet","id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken"},{"full_name":"Müller, Georg","last_name":"Müller","first_name":"Georg"},{"first_name":"Sebastian","last_name":"Peitz","orcid":"0000-0002-3389-793X","full_name":"Peitz, Sebastian","id":"47427"},{"id":"56399","full_name":"Sonntag, Konstantin","last_name":"Sonntag","orcid":"https://orcid.org/0000-0003-3384-3496","first_name":"Konstantin"},{"full_name":"Volkwein, Stefan","last_name":"Volkwein","first_name":"Stefan"}],"date_created":"2023-08-21T05:50:12Z"},{"status":"public","abstract":[{"lang":"eng","text":"Regularization is used in many different areas of optimization when solutions\r\nare sought which not only minimize a given function, but also possess a certain\r\ndegree of regularity. Popular applications are image denoising, sparse\r\nregression and machine learning. Since the choice of the regularization\r\nparameter is crucial but often difficult, path-following methods are used to\r\napproximate the entire regularization path, i.e., the set of all possible\r\nsolutions for all regularization parameters. Due to their nature, the\r\ndevelopment of these methods requires structural results about the\r\nregularization path. The goal of this article is to derive these results for\r\nthe case of a smooth objective function which is penalized by a piecewise\r\ndifferentiable regularization term. We do this by treating regularization as a\r\nmultiobjective optimization problem. Our results suggest that even in this\r\ngeneral case, the regularization path is piecewise smooth. Moreover, our theory\r\nallows for a classification of the nonsmooth features that occur in between\r\nsmooth parts. This is demonstrated in two applications, namely support-vector\r\nmachines and exact penalty methods."}],"type":"journal_article","publication":"Journal of Global Optimization","language":[{"iso":"eng"}],"user_id":"47427","department":[{"_id":"101"},{"_id":"655"}],"_id":"27426","citation":{"ama":"Gebken B, Bieker K, Peitz S. On the structure of regularization paths for piecewise differentiable regularization terms. <i>Journal of Global Optimization</i>. 2023;85(3):709-741. doi:<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>","chicago":"Gebken, Bennet, Katharina Bieker, and Sebastian Peitz. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” <i>Journal of Global Optimization</i> 85, no. 3 (2023): 709–41. <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">https://doi.org/10.1007/s10898-022-01223-2</a>.","ieee":"B. Gebken, K. Bieker, and S. Peitz, “On the structure of regularization paths for piecewise differentiable regularization terms,” <i>Journal of Global Optimization</i>, vol. 85, no. 3, pp. 709–741, 2023, doi: <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>.","bibtex":"@article{Gebken_Bieker_Peitz_2023, title={On the structure of regularization paths for piecewise differentiable regularization terms}, volume={85}, DOI={<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>}, number={3}, journal={Journal of Global Optimization}, author={Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}, year={2023}, pages={709–741} }","short":"B. Gebken, K. Bieker, S. Peitz, Journal of Global Optimization 85 (2023) 709–741.","mla":"Gebken, Bennet, et al. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” <i>Journal of Global Optimization</i>, vol. 85, no. 3, 2023, pp. 709–41, doi:<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>.","apa":"Gebken, B., Bieker, K., &#38; Peitz, S. (2023). On the structure of regularization paths for piecewise differentiable regularization terms. <i>Journal of Global Optimization</i>, <i>85</i>(3), 709–741. <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">https://doi.org/10.1007/s10898-022-01223-2</a>"},"intvolume":"        85","page":"709-741","year":"2023","issue":"3","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10898-022-01223-2.pdf"}],"doi":"10.1007/s10898-022-01223-2","title":"On the structure of regularization paths for piecewise differentiable regularization terms","date_created":"2021-11-15T09:24:59Z","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"full_name":"Bieker, Katharina","id":"32829","last_name":"Bieker","first_name":"Katharina"},{"full_name":"Peitz, Sebastian","id":"47427","last_name":"Peitz","orcid":"0000-0002-3389-793X","first_name":"Sebastian"}],"volume":85,"date_updated":"2023-03-11T17:16:33Z","oa":"1"},{"publication_identifier":{"isbn":["978-3-030-79392-0"]},"page":"43-76","citation":{"chicago":"Banholzer, Stefan, Bennet Gebken, Michael Dellnitz, Sebastian Peitz, and Stefan Volkwein. “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation.” In <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, edited by Hintermüller Michael, Herzog Roland, Kanzow Christian, Ulbrich Michael, and Ulbrich Stefan, 43–76. Cham: Springer, 2022. <a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">https://doi.org/10.1007/978-3-030-79393-7_3</a>.","ieee":"S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, and S. Volkwein, “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation,” in <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, H. Michael, H. Roland, K. Christian, U. Michael, and U. Stefan, Eds. Cham: Springer, 2022, pp. 43–76.","ama":"Banholzer S, Gebken B, Dellnitz M, Peitz S, Volkwein S. ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation. In: Michael H, Roland H, Christian K, Michael U, Stefan U, eds. <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>. Springer; 2022:43-76. doi:<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>","apa":"Banholzer, S., Gebken, B., Dellnitz, M., Peitz, S., &#38; Volkwein, S. (2022). ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation. In H. Michael, H. Roland, K. Christian, U. Michael, &#38; U. Stefan (Eds.), <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i> (pp. 43–76). Springer. <a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">https://doi.org/10.1007/978-3-030-79393-7_3</a>","short":"S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein, in: H. Michael, H. Roland, K. Christian, U. Michael, U. Stefan (Eds.), Non-Smooth and Complementarity-Based Distributed Parameter Systems, Springer, Cham, 2022, pp. 43–76.","mla":"Banholzer, Stefan, et al. “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation.” <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, edited by Hintermüller Michael et al., Springer, 2022, pp. 43–76, doi:<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>.","bibtex":"@inbook{Banholzer_Gebken_Dellnitz_Peitz_Volkwein_2022, place={Cham}, title={ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation}, DOI={<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>}, booktitle={Non-Smooth and Complementarity-Based Distributed Parameter Systems}, publisher={Springer}, author={Banholzer, Stefan and Gebken, Bennet and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}, editor={Michael, Hintermüller and Roland, Herzog and Christian, Kanzow and Michael, Ulbrich and Stefan, Ulbrich}, year={2022}, pages={43–76} }"},"place":"Cham","year":"2022","date_created":"2020-03-13T12:45:31Z","author":[{"first_name":"Stefan","last_name":"Banholzer","full_name":"Banholzer, Stefan"},{"last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643","first_name":"Bennet"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"full_name":"Peitz, Sebastian","id":"47427","last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","first_name":"Sebastian"},{"first_name":"Stefan","full_name":"Volkwein, Stefan","last_name":"Volkwein"}],"oa":"1","date_updated":"2022-03-14T13:04:51Z","publisher":"Springer","doi":"10.1007/978-3-030-79393-7_3","main_file_link":[{"url":"https://arxiv.org/pdf/1906.09075.pdf","open_access":"1"}],"title":"ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation","publication":"Non-Smooth and Complementarity-Based Distributed Parameter Systems","type":"book_chapter","status":"public","abstract":[{"text":"Multiobjective optimization plays an increasingly important role in modern\r\napplications, where several objectives are often of equal importance. The task\r\nin multiobjective optimization and multiobjective optimal control is therefore\r\nto compute the set of optimal compromises (the Pareto set) between the\r\nconflicting objectives. Since the Pareto set generally consists of an infinite\r\nnumber of solutions, the computational effort can quickly become challenging\r\nwhich is particularly problematic when the objectives are costly to evaluate as\r\nis the case for models governed by partial differential equations (PDEs). To\r\ndecrease the numerical effort to an affordable amount, surrogate models can be\r\nused to replace the expensive PDE evaluations. Existing multiobjective\r\noptimization methods using model reduction are limited either to low parameter\r\ndimensions or to few (ideally two) objectives. In this article, we present a\r\ncombination of the reduced basis model reduction method with a continuation\r\napproach using inexact gradients. The resulting approach can handle an\r\narbitrary number of objectives while yielding a significant reduction in\r\ncomputing time.","lang":"eng"}],"editor":[{"full_name":"Michael, Hintermüller","last_name":"Michael","first_name":"Hintermüller"},{"first_name":"Herzog","last_name":"Roland","full_name":"Roland, Herzog"},{"first_name":"Kanzow","full_name":"Christian, Kanzow","last_name":"Christian"},{"first_name":"Ulbrich","last_name":"Michael","full_name":"Michael, Ulbrich"},{"last_name":"Stefan","full_name":"Stefan, Ulbrich","first_name":"Ulbrich"}],"department":[{"_id":"101"},{"_id":"655"}],"user_id":"47427","_id":"16296","language":[{"iso":"eng"}]},{"external_id":{"arxiv":["2210.04579"]},"_id":"34618","department":[{"_id":"101"}],"user_id":"32643","language":[{"iso":"eng"}],"publication":"arXiv:2210.04579","type":"preprint","abstract":[{"lang":"eng","text":"In this article, we show how second-order derivative information can be\r\nincorporated into gradient sampling methods for nonsmooth optimization. The\r\nsecond-order information we consider is essentially the set of coefficients of\r\nall second-order Taylor expansions of the objective in a closed ball around a\r\ngiven point. Based on this concept, we define a model of the objective as the\r\nmaximum of these Taylor expansions. Iteratively minimizing this model\r\n(constrained to the closed ball) results in a simple descent method, for which\r\nwe prove convergence to minimal points in case the objective is convex. To\r\nobtain an implementable method, we construct an approximation scheme for the\r\nsecond-order information based on sampling objective values, gradients and\r\nHessian matrices at finitely many points. Using a set of test problems, we\r\ncompare the resulting method to five other available solvers. Considering the\r\nnumber of function evaluations, the results suggest that the method we propose\r\nis superior to the standard gradient sampling method, and competitive compared\r\nto other methods."}],"status":"public","oa":"1","date_updated":"2022-12-20T15:28:54Z","author":[{"last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643","first_name":"Bennet"}],"date_created":"2022-12-20T15:25:17Z","title":"Using second-order information in gradient sampling methods for  nonsmooth optimization","main_file_link":[{"open_access":"1","url":"https://arxiv.org/pdf/2210.04579"}],"year":"2022","citation":{"chicago":"Gebken, Bennet. “Using Second-Order Information in Gradient Sampling Methods for  Nonsmooth Optimization.” <i>ArXiv:2210.04579</i>, 2022.","ieee":"B. Gebken, “Using second-order information in gradient sampling methods for  nonsmooth optimization,” <i>arXiv:2210.04579</i>. 2022.","ama":"Gebken B. Using second-order information in gradient sampling methods for  nonsmooth optimization. <i>arXiv:221004579</i>. Published online 2022.","bibtex":"@article{Gebken_2022, title={Using second-order information in gradient sampling methods for  nonsmooth optimization}, journal={arXiv:2210.04579}, author={Gebken, Bennet}, year={2022} }","mla":"Gebken, Bennet. “Using Second-Order Information in Gradient Sampling Methods for  Nonsmooth Optimization.” <i>ArXiv:2210.04579</i>, 2022.","short":"B. Gebken, ArXiv:2210.04579 (2022).","apa":"Gebken, B. (2022). Using second-order information in gradient sampling methods for  nonsmooth optimization. In <i>arXiv:2210.04579</i>."}},{"year":"2022","citation":{"chicago":"Gebken, Bennet. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>, 2022. <a href=\"https://doi.org/10.17619/UNIPB/1-1327\">https://doi.org/10.17619/UNIPB/1-1327</a>.","ieee":"B. Gebken, <i>Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization</i>. 2022.","ama":"Gebken B. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>.; 2022. doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>","apa":"Gebken, B. (2022). <i>Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization</i>. <a href=\"https://doi.org/10.17619/UNIPB/1-1327\">https://doi.org/10.17619/UNIPB/1-1327</a>","mla":"Gebken, Bennet. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>. 2022, doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>.","bibtex":"@book{Gebken_2022, title={Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization}, DOI={<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>}, author={Gebken, Bennet}, year={2022} }","short":"B. Gebken, Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization, 2022."},"date_updated":"2022-06-01T07:13:09Z","oa":"1","supervisor":[{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"date_created":"2022-06-01T06:48:08Z","author":[{"first_name":"Bennet","id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken"}],"title":"Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization","main_file_link":[{"open_access":"1","url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/6531779"}],"doi":"10.17619/UNIPB/1-1327","type":"dissertation","abstract":[{"text":"Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind.","lang":"ger"},{"text":"Multiobjective optimization is concerned with the simultaneous optimization of multiple scalar-valued functions. In this case, a point is optimal if there is no other point that is at least as good in all objectives and better in at least one objective. A necessary condition for optimality can be derived based on first-order information of the objectives. The set of points that satisfy this necessary condition is called the Pareto critical set. This thesis presents new results about Pareto critical sets for smooth and nonsmooth multiobjective optimization problems, both in terms of their efficient computation and structural properties. In the smooth case they are computed via a continuation method and in the nonsmooth case via a descent method. Afterwards, the structure of the boundary of the Pareto critical set is analyzed, which consists of Pareto critical sets of smaller subproblems. Finally, inverse problems are considered, where a data set is given and an objective vector is sought for which the data points are critical.","lang":"eng"}],"status":"public","_id":"31556","user_id":"32643","department":[{"_id":"101"}],"language":[{"iso":"eng"}]},{"ddc":["510"],"language":[{"iso":"eng"}],"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"}],"file":[{"success":1,"relation":"main_file","content_type":"application/pdf","file_size":7990831,"file_id":"25040","access_level":"closed","file_name":"On_the_Treatment_of_Optimization_Problems_with_L1_Penalty_Terms_via_Multiobjective_Continuation.pdf","date_updated":"2021-09-25T11:59:15Z","date_created":"2021-09-25T11:59:15Z","creator":"speitz"}],"publication":"IEEE Transactions on Pattern Analysis and Machine Intelligence","title":"On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation","publisher":"IEEE","date_created":"2020-12-15T07:46:36Z","year":"2022","issue":"11","article_type":"original","file_date_updated":"2021-09-25T11:59:15Z","_id":"20731","user_id":"47427","department":[{"_id":"101"},{"_id":"530"},{"_id":"655"}],"status":"public","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9547772"}],"doi":"10.1109/TPAMI.2021.3114962","date_updated":"2022-10-21T12:27:16Z","oa":"1","author":[{"first_name":"Katharina","id":"32829","full_name":"Bieker, Katharina","last_name":"Bieker"},{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"last_name":"Peitz","orcid":"0000-0002-3389-793X","full_name":"Peitz, Sebastian","id":"47427","first_name":"Sebastian"}],"volume":44,"citation":{"short":"K. Bieker, B. Gebken, S. Peitz, IEEE Transactions on Pattern Analysis and Machine Intelligence 44 (2022) 7797–7808.","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>.","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>.","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>"},"intvolume":"        44","page":"7797-7808","publication_status":"epub_ahead","has_accepted_license":"1"},{"abstract":[{"text":"In this article, we present an efficient descent method for locally Lipschitz\r\ncontinuous multiobjective optimization problems (MOPs). The method is realized\r\nby combining a theoretical result regarding the computation of descent\r\ndirections for nonsmooth MOPs with a practical method to approximate the\r\nsubdifferentials of the objective functions. We show convergence to points\r\nwhich satisfy a necessary condition for Pareto optimality. Using a set of test\r\nproblems, we compare our method to the multiobjective proximal bundle method by\r\nM\\\"akel\\\"a. The results indicate that our method is competitive while being\r\neasier to implement. While the number of objective function evaluations is\r\nlarger, the overall number of subgradient evaluations is lower. Finally, we\r\nshow that our method can be combined with a subdivision algorithm to compute\r\nentire Pareto sets of nonsmooth MOPs.","lang":"eng"}],"status":"public","type":"journal_article","publication":"Journal of Optimization Theory and Applications","language":[{"iso":"eng"}],"_id":"16867","user_id":"47427","department":[{"_id":"101"}],"year":"2021","citation":{"apa":"Gebken, B., &#38; Peitz, S. (2021). An efficient descent method for locally Lipschitz multiobjective optimization problems. <i>Journal of Optimization Theory and Applications</i>, <i>188</i>, 696–723. <a href=\"https://doi.org/10.1007/s10957-020-01803-w\">https://doi.org/10.1007/s10957-020-01803-w</a>","bibtex":"@article{Gebken_Peitz_2021, title={An efficient descent method for locally Lipschitz multiobjective optimization problems}, volume={188}, DOI={<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>}, journal={Journal of Optimization Theory and Applications}, author={Gebken, Bennet and Peitz, Sebastian}, year={2021}, pages={696–723} }","mla":"Gebken, Bennet, and Sebastian Peitz. “An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems.” <i>Journal of Optimization Theory and Applications</i>, vol. 188, 2021, pp. 696–723, doi:<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>.","short":"B. Gebken, S. Peitz, Journal of Optimization Theory and Applications 188 (2021) 696–723.","ieee":"B. Gebken and S. Peitz, “An efficient descent method for locally Lipschitz multiobjective optimization problems,” <i>Journal of Optimization Theory and Applications</i>, vol. 188, pp. 696–723, 2021.","chicago":"Gebken, Bennet, and Sebastian Peitz. “An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems.” <i>Journal of Optimization Theory and Applications</i> 188 (2021): 696–723. <a href=\"https://doi.org/10.1007/s10957-020-01803-w\">https://doi.org/10.1007/s10957-020-01803-w</a>.","ama":"Gebken B, Peitz S. An efficient descent method for locally Lipschitz multiobjective optimization problems. <i>Journal of Optimization Theory and Applications</i>. 2021;188:696-723. doi:<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>"},"intvolume":"       188","page":"696-723","publication_status":"published","title":"An efficient descent method for locally Lipschitz multiobjective optimization problems","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10957-020-01803-w.pdf"}],"doi":"10.1007/s10957-020-01803-w","date_updated":"2022-01-06T06:52:57Z","oa":"1","author":[{"first_name":"Bennet","full_name":"Gebken, Bennet","id":"32643","last_name":"Gebken"},{"id":"47427","full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz","first_name":"Sebastian"}],"date_created":"2020-04-27T09:11:22Z","volume":188},{"publication":"Journal of Global Optimization","type":"journal_article","abstract":[{"text":"It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.","lang":"eng"}],"status":"public","_id":"16295","department":[{"_id":"101"}],"user_id":"47427","language":[{"iso":"eng"}],"year":"2021","page":"3-29","intvolume":"        80","citation":{"apa":"Gebken, B., &#38; Peitz, S. (2021). Inverse multiobjective optimization: Inferring decision criteria from data. <i>Journal of Global Optimization</i>, <i>80</i>, 3–29. <a href=\"https://doi.org/10.1007/s10898-020-00983-z\">https://doi.org/10.1007/s10898-020-00983-z</a>","short":"B. Gebken, S. Peitz, Journal of Global Optimization 80 (2021) 3–29.","mla":"Gebken, Bennet, and Sebastian Peitz. “Inverse Multiobjective Optimization: Inferring Decision Criteria from Data.” <i>Journal of Global Optimization</i>, vol. 80, Springer, 2021, pp. 3–29, doi:<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>.","bibtex":"@article{Gebken_Peitz_2021, title={Inverse multiobjective optimization: Inferring decision criteria from data}, volume={80}, DOI={<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>}, journal={Journal of Global Optimization}, publisher={Springer}, author={Gebken, Bennet and Peitz, Sebastian}, year={2021}, pages={3–29} }","ieee":"B. Gebken and S. Peitz, “Inverse multiobjective optimization: Inferring decision criteria from data,” <i>Journal of Global Optimization</i>, vol. 80, pp. 3–29, 2021.","chicago":"Gebken, Bennet, and Sebastian Peitz. “Inverse Multiobjective Optimization: Inferring Decision Criteria from Data.” <i>Journal of Global Optimization</i> 80 (2021): 3–29. <a href=\"https://doi.org/10.1007/s10898-020-00983-z\">https://doi.org/10.1007/s10898-020-00983-z</a>.","ama":"Gebken B, Peitz S. Inverse multiobjective optimization: Inferring decision criteria from data. <i>Journal of Global Optimization</i>. 2021;80:3-29. doi:<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>"},"oa":"1","publisher":"Springer","date_updated":"2022-01-06T06:52:48Z","volume":80,"date_created":"2020-03-13T12:45:05Z","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"full_name":"Peitz, Sebastian","id":"47427","last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","first_name":"Sebastian"}],"title":"Inverse multiobjective optimization: Inferring decision criteria from data","doi":"10.1007/s10898-020-00983-z","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10898-020-00983-z.pdf"}]},{"language":[{"iso":"eng"}],"department":[{"_id":"101"}],"user_id":"32655","_id":"16712","status":"public","abstract":[{"lang":"eng","text":"We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions."}],"publication":"Dynamical Systems","type":"journal_article","doi":"10.1080/14689367.2019.1661355","main_file_link":[{"url":"https://doi.org/10.1080/14689367.2019.1661355"}],"title":"On the equivariance properties of self-adjoint matrices","volume":35,"author":[{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"first_name":"Bennet","full_name":"Gebken, Bennet","id":"32643","last_name":"Gebken"},{"last_name":"Gerlach","id":"32655","full_name":"Gerlach, Raphael","first_name":"Raphael"},{"last_name":"Klus","full_name":"Klus, Stefan","first_name":"Stefan"}],"date_created":"2020-04-16T14:07:25Z","date_updated":"2023-11-17T13:12:59Z","intvolume":"        35","page":"197-215","citation":{"chicago":"Dellnitz, Michael, Bennet Gebken, Raphael Gerlach, and Stefan Klus. “On the Equivariance Properties of Self-Adjoint Matrices.” <i>Dynamical Systems</i> 35, no. 2 (2020): 197–215. <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">https://doi.org/10.1080/14689367.2019.1661355</a>.","ieee":"M. Dellnitz, B. Gebken, R. Gerlach, and S. Klus, “On the equivariance properties of self-adjoint matrices,” <i>Dynamical Systems</i>, vol. 35, no. 2, pp. 197–215, 2020, doi: <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>.","ama":"Dellnitz M, Gebken B, Gerlach R, Klus S. On the equivariance properties of self-adjoint matrices. <i>Dynamical Systems</i>. 2020;35(2):197-215. doi:<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>","short":"M. Dellnitz, B. Gebken, R. Gerlach, S. Klus, Dynamical Systems 35 (2020) 197–215.","mla":"Dellnitz, Michael, et al. “On the Equivariance Properties of Self-Adjoint Matrices.” <i>Dynamical Systems</i>, vol. 35, no. 2, 2020, pp. 197–215, doi:<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>.","bibtex":"@article{Dellnitz_Gebken_Gerlach_Klus_2020, title={On the equivariance properties of self-adjoint matrices}, volume={35}, DOI={<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>}, number={2}, journal={Dynamical Systems}, author={Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}, year={2020}, pages={197–215} }","apa":"Dellnitz, M., Gebken, B., Gerlach, R., &#38; Klus, S. (2020). On the equivariance properties of self-adjoint matrices. <i>Dynamical Systems</i>, <i>35</i>(2), 197–215. <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">https://doi.org/10.1080/14689367.2019.1661355</a>"},"year":"2020","issue":"2","publication_identifier":{"issn":["1468-9367","1468-9375"]},"publication_status":"published"},{"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set."}],"publication":"Journal of Global Optimization","title":"On the hierarchical structure of Pareto critical sets","date_created":"2019-07-10T08:13:31Z","year":"2019","issue":"4","article_type":"original","user_id":"47427","department":[{"_id":"101"}],"_id":"10595","status":"public","type":"journal_article","doi":"10.1007/s10898-019-00737-6","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"first_name":"Sebastian","last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","full_name":"Peitz, Sebastian","id":"47427"},{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"volume":73,"date_updated":"2022-01-06T06:50:46Z","citation":{"short":"B. Gebken, S. Peitz, M. Dellnitz, Journal of Global Optimization 73 (2019) 891–913.","bibtex":"@article{Gebken_Peitz_Dellnitz_2019, title={On the hierarchical structure of Pareto critical sets}, volume={73}, DOI={<a href=\"https://doi.org/10.1007/s10898-019-00737-6\">10.1007/s10898-019-00737-6</a>}, number={4}, journal={Journal of Global Optimization}, author={Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael}, year={2019}, pages={891–913} }","mla":"Gebken, Bennet, et al. “On the Hierarchical Structure of Pareto Critical Sets.” <i>Journal of Global Optimization</i>, vol. 73, no. 4, 2019, pp. 891–913, doi:<a href=\"https://doi.org/10.1007/s10898-019-00737-6\">10.1007/s10898-019-00737-6</a>.","apa":"Gebken, B., Peitz, S., &#38; Dellnitz, M. (2019). On the hierarchical structure of Pareto critical sets. <i>Journal of Global Optimization</i>, <i>73</i>(4), 891–913. <a href=\"https://doi.org/10.1007/s10898-019-00737-6\">https://doi.org/10.1007/s10898-019-00737-6</a>","ieee":"B. Gebken, S. Peitz, and M. Dellnitz, “On the hierarchical structure of Pareto critical sets,” <i>Journal of Global Optimization</i>, vol. 73, no. 4, pp. 891–913, 2019.","chicago":"Gebken, Bennet, Sebastian Peitz, and Michael Dellnitz. “On the Hierarchical Structure of Pareto Critical Sets.” <i>Journal of Global Optimization</i> 73, no. 4 (2019): 891–913. <a href=\"https://doi.org/10.1007/s10898-019-00737-6\">https://doi.org/10.1007/s10898-019-00737-6</a>.","ama":"Gebken B, Peitz S, Dellnitz M. On the hierarchical structure of Pareto critical sets. <i>Journal of Global Optimization</i>. 2019;73(4):891-913. doi:<a href=\"https://doi.org/10.1007/s10898-019-00737-6\">10.1007/s10898-019-00737-6</a>"},"intvolume":"        73","page":"891-913","publication_status":"published","publication_identifier":{"issn":["0925-5001","1573-2916"]}},{"publication_identifier":{"isbn":["9783319961033","9783319961040"],"issn":["1860-949X","1860-9503"]},"publication_status":"published","place":"Cham","year":"2018","citation":{"mla":"Gebken, Bennet, et al. “A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems.” <i>Numerical and Evolutionary Optimization – NEO 2017</i>, 2018, doi:<a href=\"https://doi.org/10.1007/978-3-319-96104-0_2\">10.1007/978-3-319-96104-0_2</a>.","short":"B. Gebken, S. Peitz, M. Dellnitz, in: Numerical and Evolutionary Optimization – NEO 2017, Cham, 2018.","bibtex":"@inproceedings{Gebken_Peitz_Dellnitz_2018, place={Cham}, title={A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems}, DOI={<a href=\"https://doi.org/10.1007/978-3-319-96104-0_2\">10.1007/978-3-319-96104-0_2</a>}, booktitle={Numerical and Evolutionary Optimization – NEO 2017}, author={Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael}, year={2018} }","apa":"Gebken, B., Peitz, S., &#38; Dellnitz, M. (2018). A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems. In <i>Numerical and Evolutionary Optimization – NEO 2017</i>. Cham. <a href=\"https://doi.org/10.1007/978-3-319-96104-0_2\">https://doi.org/10.1007/978-3-319-96104-0_2</a>","chicago":"Gebken, Bennet, Sebastian Peitz, and Michael Dellnitz. “A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems.” In <i>Numerical and Evolutionary Optimization – NEO 2017</i>. Cham, 2018. <a href=\"https://doi.org/10.1007/978-3-319-96104-0_2\">https://doi.org/10.1007/978-3-319-96104-0_2</a>.","ieee":"B. Gebken, S. Peitz, and M. Dellnitz, “A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems,” in <i>Numerical and Evolutionary Optimization – NEO 2017</i>, 2018.","ama":"Gebken B, Peitz S, Dellnitz M. A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems. In: <i>Numerical and Evolutionary Optimization – NEO 2017</i>. Cham; 2018. doi:<a href=\"https://doi.org/10.1007/978-3-319-96104-0_2\">10.1007/978-3-319-96104-0_2</a>"},"date_updated":"2022-01-06T07:04:00Z","date_created":"2019-03-29T13:26:47Z","author":[{"first_name":"Bennet","full_name":"Gebken, Bennet","id":"32643","last_name":"Gebken"},{"orcid":"https://orcid.org/0000-0002-3389-793X","last_name":"Peitz","full_name":"Peitz, Sebastian","id":"47427","first_name":"Sebastian"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"}],"title":"A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems","conference":{"name":"NEO 2017: Numerical and Evolutionary Optimization"},"doi":"10.1007/978-3-319-96104-0_2","publication":"Numerical and Evolutionary Optimization – NEO 2017","type":"conference","abstract":[{"text":"In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.","lang":"eng"}],"status":"public","_id":"8750","department":[{"_id":"101"}],"user_id":"47427","language":[{"iso":"eng"}]}]