{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T05:02:27Z","timestamp":1774414947681,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:00:00Z","timestamp":1723075200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:00:00Z","timestamp":1723075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100020595","name":"National Science and Technology Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100020595","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s10107-024-02113-z","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T06:03:01Z","timestamp":1723096981000},"page":"479-518","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Unified smoothing approach for best hyperparameter selection problem using a bilevel optimization strategy"],"prefix":"10.1007","volume":"212","author":[{"given":"Jan Harold","family":"Alcantara","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chieu Thanh","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takayuki","family":"Okuno","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Akiko","family":"Takeda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-9419","authenticated-orcid":false,"given":"Jein-Shan","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,8]]},"reference":[{"issue":"2","key":"2113_CR1","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.orl.2020.01.002","volume":"48","author":"JH Alcantara","year":"2020","unstructured":"Alcantara, J.H., Lee, C.H., Nguyen, C.T., Chang, Y.L., Chen, J.-S.: On construction of new NCP functions. Oper. Res. Lett. 48(2), 115\u2013121 (2020)","journal-title":"Oper. Res. Lett."},{"issue":"10","key":"2113_CR2","first-page":"281","volume":"13","author":"J Bergstra","year":"2012","unstructured":"Bergstra, J., Bengio, Y.: Random search for hyper-parameter optimization. J. Mach. Learn. Res. 13(10), 281\u2013305 (2012)","journal-title":"J. Mach. Learn. Res."},{"key":"2113_CR3","doi-asserted-by":"crossref","unstructured":"Bennett, K.P., Hu, J., Ji, X., Kunapuli, G., Pang, J-S.: Model selection via bilevel optimization. In The 2006 IEEE International Joint Conference on Neural Network Proceedings, pp. 1922\u20131929 (2006)","DOI":"10.1109\/IJCNN.2006.246935"},{"key":"2113_CR4","unstructured":"Bennett, K.P., Hu, J., Ji, X., Kunapuli, G., Pang, J-S.: Bilevel optimization and machine learning. In: Computational Intelligence: Research Frontiers. WCCI 2008. Lecture Notes in Computer Science, vol. 5050. Springer, Berlin, Heidelberg (2008)"},{"issue":"1","key":"2113_CR5","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1287\/opre.21.1.37","volume":"21","author":"J Bracken","year":"1973","unstructured":"Bracken, J., McGill, J.: Mathematical programs with optimization problems in the constraints. Oper. Res. 21(1), 37\u201344 (1973)","journal-title":"Oper. Res."},{"issue":"1","key":"2113_CR6","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s10107-012-0569-0","volume":"134","author":"X Chen","year":"2012","unstructured":"Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134(1), 71\u201399 (2012)","journal-title":"Math. Program."},{"issue":"2","key":"2113_CR7","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF00249052","volume":"5","author":"C Chen","year":"1996","unstructured":"Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5(2), 97\u2013138 (1996)","journal-title":"Comput. Optim. Appl."},{"issue":"5","key":"2113_CR8","doi-asserted-by":"publisher","first-page":"2832","DOI":"10.1137\/090761471","volume":"32","author":"X Chen","year":"2010","unstructured":"Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of $$\\ell _2$$-$$\\ell _p$$ minimization. SIAM J. Sci. Comput. 32(5), 2832\u20132852 (2010)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2113_CR9","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1137\/080740167","volume":"3","author":"X Chen","year":"2010","unstructured":"Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 3(4), 765\u2013790 (2010)","journal-title":"SIAM J. Imaging Sci."},{"key":"2113_CR10","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1007\/s10479-007-0176-2","volume":"153","author":"B Colson","year":"2007","unstructured":"Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 234\u2013256 (2007)","journal-title":"Ann. Oper. Res."},{"issue":"3","key":"2113_CR11","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1080\/0233193031000149894","volume":"52","author":"S Dempe","year":"2003","unstructured":"Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333\u2013359 (2003)","journal-title":"Optimization"},{"issue":"3","key":"2113_CR12","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/BF02592192","volume":"75","author":"T De Luca","year":"1996","unstructured":"De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75(3), 407\u2013439 (1996)","journal-title":"Math. Program."},{"key":"2113_CR13","volume-title":"Finite Dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.-S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. 1 and 2. Springer, New York (2003)"},{"issue":"2","key":"2113_CR14","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1214\/009053607000000875","volume":"36","author":"J Huang","year":"2008","unstructured":"Huang, J., Horowitz, J.L., Ma, S.: Asymptotic properties of bridge estimators in sparse high-dimensional regression models. Ann. Statist. 36(2), 587\u2013613 (2008)","journal-title":"Ann. Statist."},{"issue":"3","key":"2113_CR15","doi-asserted-by":"publisher","first-page":"3325","DOI":"10.1023\/A:1023673105317","volume":"116","author":"AY Kruger","year":"2003","unstructured":"Kruger, A.Y.: On Fr\u00e9chet subdifferentials. J. Math. Sci. 116(3), 3325\u20133358 (2003)","journal-title":"J. Math. Sci."},{"issue":"2","key":"2113_CR16","doi-asserted-by":"publisher","first-page":"938","DOI":"10.1137\/120882706","volume":"6","author":"K Kunisch","year":"2013","unstructured":"Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6(2), 938\u2013983 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"2113_CR17","doi-asserted-by":"crossref","unstructured":"Moore, G.M., Bergeron, C., Bennett, K.P.: Nonsmooth bilevel programming for hyperparameter selection. In IEEE International Conference on Data Mining Workshops, pp. 374\u2013381 (2009)","DOI":"10.1109\/ICDMW.2009.74"},{"key":"2113_CR18","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s10994-011-5246-7","volume":"85","author":"GM Moore","year":"2011","unstructured":"Moore, G.M., Bergeron, C., Bennett, K.P.: Model selection for primal SVM. Mach. Learn. 85, 175\u2013208 (2011)","journal-title":"Mach. Learn."},{"issue":"1","key":"2113_CR19","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1137\/070692285","volume":"1","author":"M Nikolova","year":"2008","unstructured":"Nikolova, M., Ng, M.K., Zhang, S., Ching, W.-K.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1(1), 2\u201325 (2008)","journal-title":"SIAM J. Imaging Sci."},{"issue":"2","key":"2113_CR20","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s10851-016-0663-7","volume":"56","author":"P Ochs","year":"2016","unstructured":"Ochs, P., Ranftl, R., Brox, T., Pock, T.: Techniques for gradient-based bilevel optimization with non-smooth lower level problems. J. Math. Imaging Vis. 56(2), 175\u2013194 (2016)","journal-title":"J. Math. Imaging Vis."},{"issue":"245","key":"2113_CR21","first-page":"1","volume":"22","author":"T Okuno","year":"2021","unstructured":"Okuno, T., Takeda, A., Kawana, A., Watanabe, M.: On $$\\ell _p$$-hyperparameter learning via bilevel nonsmooth optimization. J. Mach. Learn. Res. 22(245), 1\u201347 (2021)","journal-title":"J. Mach. Learn. Res."},{"key":"2113_CR22","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer, Cham (2009)"},{"key":"2113_CR23","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s12190-019-01262-1","volume":"61","author":"B Saheya","year":"2019","unstructured":"Saheya, B., Nguyen, C.T., Chen, J.-S.: Neural network based on systematically generated smoothing functions for absolute value equation. J. Appl. Math. Comput. 61, 533\u2013558 (2019)","journal-title":"J. Appl. Math. Comput."},{"issue":"2","key":"2113_CR24","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1109\/TEVC.2017.2712906","volume":"22","author":"A Sinha","year":"2018","unstructured":"Sinha, A., Malo, P., Deb, K.: A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE Trans. Evol. Comput. 22(2), 276\u2013295 (2018)","journal-title":"IEEE Trans. Evol. Comput."},{"issue":"1","key":"2113_CR25","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Statist. Soc. Ser. B. 58(1), 267\u2013288 (1996)","journal-title":"J. R. Statist. Soc. Ser. B."},{"key":"2113_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.compchemeng.2020.106902","volume":"139","author":"WW Tso","year":"2020","unstructured":"Tso, W.W., Burnak, B., Pistikopoulos, E.N.: HY-POP: hyperparameter optimization of machine learning models through parametric programming. Comput. Chem. Eng. 139, 106902 (2020)","journal-title":"Comput. Chem. Eng."},{"key":"2113_CR27","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1613\/jair.4806","volume":"55","author":"Z Wang","year":"2016","unstructured":"Wang, Z., Hutter, F., Zoghi, M., Matheson, D., de Freitas, N.: Bayesian optimization in a billion dimensions via random embeddings. J. Artif. Intell. Res. 55, 361\u2013387 (2016)","journal-title":"J. Artif. Intell. Res."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02113-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-024-02113-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02113-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T16:03:12Z","timestamp":1750176192000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-024-02113-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,8]]},"references-count":27,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["2113"],"URL":"https:\/\/doi.org\/10.1007\/s10107-024-02113-z","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,8]]},"assertion":[{"value":"21 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}