{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T07:08:08Z","timestamp":1780988888617,"version":"3.54.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T00:00:00Z","timestamp":1771545600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T00:00:00Z","timestamp":1771545600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt Regular 1240120"],"award-info":[{"award-number":["Fondecyt Regular 1240120"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt Regular 1240120"],"award-info":[{"award-number":["Fondecyt Regular 1240120"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt Regular 1220886"],"award-info":[{"award-number":["Fondecyt Regular 1220886"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt Regular 1220886"],"award-info":[{"award-number":["Fondecyt Regular 1220886"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Fondecyt Regular 1240335"],"award-info":[{"award-number":["Fondecyt Regular 1240335"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Proyecto de Exploraci\u00f3n 13220097"],"award-info":[{"award-number":["Proyecto de Exploraci\u00f3n 13220097"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Proyecto de Exploraci\u00f3n 13220097"],"award-info":[{"award-number":["Proyecto de Exploraci\u00f3n 13220097"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Basal Funds FB210005"],"award-info":[{"award-number":["Basal Funds FB210005"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Basal Funds FB210005"],"award-info":[{"award-number":["Basal Funds FB210005"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["ECOS230027"],"award-info":[{"award-number":["ECOS230027"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["ECOS230027"],"award-info":[{"award-number":["ECOS230027"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["MATH-AMSUD 23-MATH-17"],"award-info":[{"award-number":["MATH-AMSUD 23-MATH-17"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["MATH-AMSUD 23-MATH-17"],"award-info":[{"award-number":["MATH-AMSUD 23-MATH-17"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["Doctorado Nacional 21210951"],"award-info":[{"award-number":["Doctorado Nacional 21210951"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s10957-025-02921-z","type":"journal-article","created":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T08:40:19Z","timestamp":1771576819000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions"],"prefix":"10.1007","volume":"208","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-6915-0793","authenticated-orcid":false,"given":"Sergio","family":"L\u00f3pez-Rivera","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pedro","family":"P\u00e9rez-Aros","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emilio","family":"Vilches","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,20]]},"reference":[{"issue":"3","key":"2921_CR1","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/s0294-1449(16)30214-1","volume":"10","author":"H Attouch","year":"1993","unstructured":"Attouch, H., Az\u00e9, D.: Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method. Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire 10(3), 289\u2013312 (1993)","journal-title":"Ann. Inst. H. Poincar\u00e9 C Anal. Non Lin\u00e9aire"},{"issue":"1","key":"2921_CR2","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., Svaiter, B.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1), 91\u2013129 (2013)","journal-title":"Math. Program."},{"key":"2921_CR3","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1007\/s11263-006-4331-z","volume":"67","author":"J Aujol","year":"2006","unstructured":"Aujol, J., Gilboa, G., Chan, T., Osher, S.: Structure-texture image decomposition - modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67, 111\u2013136 (2006)","journal-title":"Int. J. Comput. Vis."},{"key":"2921_CR4","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. CMS Books Math.\/Ouvrages Math. SMC, Springer, Berlin (2017)","DOI":"10.1007\/978-3-319-48311-5"},{"issue":"2","key":"2921_CR5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1080\/00401706.1974.10489171","volume":"16","author":"A Beaton","year":"1974","unstructured":"Beaton, A., Tukey, J.: The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16(2), 147\u2013185 (1974)","journal-title":"Technometrics"},{"key":"2921_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume-title":"First-Order Methods in Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-Order Methods in Optimization, vol. 25. SIAM, Philadelphia, PA (2017)"},{"key":"2921_CR7","doi-asserted-by":"publisher","first-page":"628","DOI":"10.1007\/s10957-020-01800-z","volume":"188","author":"A B\u00f6hm","year":"2021","unstructured":"B\u00f6hm, A., Wright, S.J.: Variable smoothing for weakly convex composite functions. J. Optim. Theory Appl. 188, 628\u2013649 (2021)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2921_CR8","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s00245-022-09921-9","volume":"87","author":"L Brice\u00f1o-Arias","year":"2023","unstructured":"Brice\u00f1o-Arias, L., Deride, J., L\u00f3pez-Rivera, S., Silva, F.: A primal-dual partial inverse algorithm for constrained monotone inclusions: applications to stochastic programming and mean field games. Appl. Math. Optim. 87(2), 21 (2023)","journal-title":"Appl. Math. Optim."},{"issue":"5","key":"2921_CR9","doi-asserted-by":"publisher","first-page":"1239","DOI":"10.1080\/02331934.2013.855210","volume":"64","author":"L Brice\u00f1o-Arias","year":"2015","unstructured":"Brice\u00f1o-Arias, L.: Forward-Douglas-Rachford splitting and forward-partial inverse method for solving monotone inclusions. Optimization 64(5), 1239\u20131261 (2015)","journal-title":"Optimization"},{"issue":"1","key":"2921_CR10","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10589-022-00395-7","volume":"83","author":"R Campoy","year":"2022","unstructured":"Campoy, R.: A product space reformulation with reduced dimension for splitting algorithms. Comput. Optim. Appl. 83(1), 319\u2013348 (2022)","journal-title":"Comput. Optim. Appl."},{"key":"2921_CR11","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s002110050258","volume":"76","author":"A Chambolle","year":"1997","unstructured":"Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167\u2013188 (1997)","journal-title":"Numer. Math."},{"issue":"1, Ser. B","key":"2921_CR12","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s10107-017-1132-9","volume":"165","author":"X Chen","year":"2017","unstructured":"Chen, X., Pong, T.K., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165(1, Ser. B), 71\u2013111 (2017)","journal-title":"Math. Program."},{"issue":"2","key":"2921_CR13","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1007\/s10957-012-0245-9","volume":"158","author":"L Condat","year":"2013","unstructured":"Condat, L.: A primal-dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460\u2013479 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"2921_CR14","volume-title":"Modern Nonconvex Nondifferentiable Optimization","author":"Y Cui","year":"2022","unstructured":"Cui, Y., Pang, J.S.: Modern Nonconvex Nondifferentiable Optimization, vol. 29. SIAM, Philadelphia, PA (2022)"},{"issue":"6","key":"2921_CR15","doi-asserted-by":"publisher","first-page":"1385","DOI":"10.1287\/opre.28.6.1385","volume":"28","author":"B Dasarathy","year":"1980","unstructured":"Dasarathy, B., White, L.: A maxmin location problem. Oper. Res. 28(6), 1385\u20131401 (1980)","journal-title":"Oper. Res."},{"issue":"4","key":"2921_CR16","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1007\/s11228-021-00600-5","volume":"29","author":"W De Oliveira","year":"2021","unstructured":"De Oliveira, W.: Risk-averse stochastic programming and distributionally robust optimization via operator splitting. Set-Valued Var. Anal. 29(4), 861\u2013891 (2021)","journal-title":"Set-Valued Var. Anal."},{"issue":"456","key":"2921_CR17","doi-asserted-by":"publisher","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96(456), 1348\u20131360 (2001)","journal-title":"J. Amer. Statist. Assoc."},{"issue":"4","key":"2921_CR18","doi-asserted-by":"publisher","first-page":"2264","DOI":"10.1137\/120888880","volume":"23","author":"S Haines","year":"2013","unstructured":"Haines, S., Loeppky, J., Tseng, P., Wang, X.: Convex relaxations of the weighted maxmin dispersion problem. SIAM J. Optim. 23(4), 2264\u20132294 (2013)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"2921_CR19","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1007\/BF01580223","volume":"6","author":"M Held","year":"1974","unstructured":"Held, M., Wolfe, P., Crowder, H.P.: Validation of subgradient optimization. Math. Program. 6(1), 62\u201388 (1974)","journal-title":"Math. Program."},{"issue":"1","key":"2921_CR20","doi-asserted-by":"publisher","first-page":"760","DOI":"10.1137\/16M1058480","volume":"28","author":"V Jeyakumar","year":"2018","unstructured":"Jeyakumar, V., Li, G.: Exact second-order cone programming relaxations for some nonconvex minimax quadratic optimization problems. SIAM J. Optim. 28(1), 760\u2013787 (2018)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"2921_CR21","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/0378-3758(90)90122-B","volume":"26","author":"M Johnson","year":"1990","unstructured":"Johnson, M., Moore, L., Ylvisaker, D.: Minimax and maximin distance designs. J. Statist. Plann. Inference 26(2), 131\u2013148 (1990)","journal-title":"J. Statist. Plann. Inference"},{"issue":"3","key":"2921_CR22","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s10898-024-01460-7","volume":"91","author":"PD Khanh","year":"2025","unstructured":"Khanh, P.D., Mordukhovich, B.S., Phat, V.T., Tran, D.B.: Inexact proximal methods for weakly convex functions. J. Global Optim. 91(3), 611\u2013646 (2025)","journal-title":"J. Global Optim."},{"key":"2921_CR23","volume-title":"Linear and Nonlinear Programming","author":"D Luenberger","year":"1984","unstructured":"Luenberger, D., Ye, Y.: Linear and Nonlinear Programming, vol. 2. Springer, Berlin (1984)"},{"key":"2921_CR24","unstructured":"Mercier, B.: In\u00e9quations Variationnelles de la M\u00e9canique, vol.\u00a01. Universit\u00e9 de Paris-Sud, D\u00e9partement de Math\u00e9matiques, Orsay (1980)"},{"issue":"3","key":"2921_CR25","doi-asserted-by":"publisher","first-page":"1635","DOI":"10.1137\/20M1373517","volume":"31","author":"P P\u00e9rez-Aros","year":"2021","unstructured":"P\u00e9rez-Aros, P., Vilches, E.: Moreau envelope of supremum functions with applications to infinite and stochastic programming. SIAM J. Optim. 31(3), 1635\u20131657 (2021)","journal-title":"SIAM J. Optim."},{"key":"2921_CR26","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1997","unstructured":"Rockafellar, R.T.: Convex Analysis, vol. 11. Princeton University Press, Princeton (1997)"},{"key":"2921_CR27","doi-asserted-by":"crossref","unstructured":"Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (2021)","DOI":"10.1137\/1.9781611976595"},{"key":"2921_CR28","doi-asserted-by":"crossref","unstructured":"Shi, W., Wahba, G., Wright, S., Lee, K., Klein, R., Klein, B.: LASSO-patternsearch algorithm with application to ophthalmology data. Stat. Interface 1, 137\u2013153 (2008)","DOI":"10.4310\/SII.2008.v1.n1.a12"},{"key":"2921_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-85128-6","volume-title":"Robust Optimization in Electric Energy Systems","author":"X Sun","year":"2021","unstructured":"Sun, X., Conejo, A.: Robust Optimization in Electric Energy Systems. Springer, Berlin (2021)"},{"key":"2921_CR30","doi-asserted-by":"crossref","unstructured":"Tibshirani, R., Taylor, J.: The solution path of the generalized LASSO. Ann. Statist. 39(3), 1335\u20131371 (2011)","DOI":"10.1214\/11-AOS878"},{"issue":"2","key":"2921_CR31","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"CH Zhang","year":"2010","unstructured":"Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Statist."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02921-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02921-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02921-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T06:47:51Z","timestamp":1780987671000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02921-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,20]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2921"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02921-z","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,20]]},"assertion":[{"value":"1 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"115"}}