{"page":"709-741","abstract":[{"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.","lang":"eng"}],"date_created":"2021-11-15T09:24:59Z","status":"public","doi":"10.1007/s10898-022-01223-2","type":"journal_article","citation":{"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>.","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} }","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>.","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>","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>"},"issue":"3","date_updated":"2023-03-11T17:16:33Z","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10898-022-01223-2.pdf"}],"language":[{"iso":"eng"}],"publication":"Journal of Global Optimization","_id":"27426","title":"On the structure of regularization paths for piecewise differentiable regularization terms","author":[{"id":"32643","first_name":"Bennet","full_name":"Gebken, Bennet","last_name":"Gebken"},{"first_name":"Katharina","id":"32829","last_name":"Bieker","full_name":"Bieker, Katharina"},{"last_name":"Peitz","full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","id":"47427","first_name":"Sebastian"}],"year":"2023","volume":85,"oa":"1","department":[{"_id":"101"},{"_id":"655"}],"user_id":"47427","intvolume":"        85"}