{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T18:50:15Z","timestamp":1755802215141,"version":"3.44.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T00:00:00Z","timestamp":1748217600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T00:00:00Z","timestamp":1748217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["307987\/2023-0"],"award-info":[{"award-number":["307987\/2023-0"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001807","name":"FAPESP","doi-asserted-by":"crossref","award":["2022\/06745-7","2024\/15980-5"],"award-info":[{"award-number":["2022\/06745-7","2024\/15980-5"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s10589-025-00694-9","type":"journal-article","created":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T09:02:04Z","timestamp":1748250124000},"page":"327-344","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A unified approach for smoothing approximations to the exact $$\\ell _1$$-penalty for inequality-constrained optimization"],"prefix":"10.1007","volume":"92","author":[{"given":"Mariana da","family":"Rosa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9176-2999","authenticated-orcid":false,"given":"Ademir Alves","family":"Ribeiro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elizabeth Wegner","family":"Karas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,26]]},"reference":[{"key":"694_CR1","doi-asserted-by":"publisher","unstructured":"Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Classics in Applied Mathematics. SIAM, Philadelphia (1990). https:\/\/doi.org\/10.1137\/1.9781611971316","DOI":"10.1137\/1.9781611971316"},{"key":"694_CR2","doi-asserted-by":"publisher","unstructured":"Fletcher, R.: Practical Methods of Optimization, 2nd edn. John Wiley & Sons, New York (1987). https:\/\/doi.org\/10.1002\/9781118723203","DOI":"10.1002\/9781118723203"},{"issue":"5","key":"694_CR3","doi-asserted-by":"publisher","first-page":"344","DOI":"10.1287\/mnsc.13.5.344","volume":"13","author":"WI Zangwill","year":"1967","unstructured":"Zangwill, W.I.: Non-linear programming via penalty functions. Manag. Sci. 13(5), 344\u2013358 (1967). https:\/\/doi.org\/10.1287\/mnsc.13.5.344","journal-title":"Manag. Sci."},{"key":"694_CR4","doi-asserted-by":"publisher","unstructured":"Di\u00a0Pillo, G.: Exact Penalty Methods. In: Spedicato, E. (ed.) Algorithms for Continuous Optimization: The State of the Art. NATO ASI Series, 434, pp. 209\u2013253. Springer, Dordrecht (1994). https:\/\/doi.org\/10.1007\/978-94-009-0369-2_8","DOI":"10.1007\/978-94-009-0369-2_8"},{"issue":"1","key":"694_CR5","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1093\/amrx\/abq003","volume":"2010","author":"JD Griffin","year":"2010","unstructured":"Griffin, J.D., Kolda, T.G.: Nonlinearly constrained optimization using heuristic penalty methods and asynchronous parallel generating set search. Appl. Math. Res. Express 2010(1), 36\u201362 (2010). https:\/\/doi.org\/10.1093\/amrx\/abq003","journal-title":"Appl. Math. Res. Express"},{"issue":"3","key":"694_CR6","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/BF02346160","volume":"57","author":"G Di Pillo","year":"1988","unstructured":"Di Pillo, G., Grippo, L.: On the exactness of a class of nondifferentiable penalty functions. J. Optim. Theory Appl. 57(3), 399\u2013410 (1988). https:\/\/doi.org\/10.1007\/BF02346160","journal-title":"J. Optim. Theory Appl."},{"key":"694_CR7","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1007\/s40314-015-0291-7","volume":"36","author":"AA Ribeiro","year":"2017","unstructured":"Ribeiro, A.A., Sachine, M., Santos, S.A.: On the augmented subproblems within sequential methods for nonlinear programming. Comp. Appl. Math. 36, 1255\u20131272 (2017). https:\/\/doi.org\/10.1007\/s40314-015-0291-7","journal-title":"Comp. Appl. Math."},{"key":"694_CR8","doi-asserted-by":"publisher","first-page":"6601","DOI":"10.1007\/s40314-018-0702-7","volume":"37","author":"AA Ribeiro","year":"2018","unstructured":"Ribeiro, A.A., Sachine, M., Santos, S.A.: On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming. Comp. Appl. Math. 37, 6601\u20136618 (2018). https:\/\/doi.org\/10.1007\/s40314-018-0702-7","journal-title":"Comp. Appl. Math."},{"key":"694_CR9","doi-asserted-by":"publisher","unstructured":"Bertsekas, D.P.: Nondifferentiable Optimization via Approximation. In: Balinski, M.L., Wolfe, P. (eds.) Nondifferentiable Optimization. Mathematical Programming Studies, vol. 3. Springer, Berlin, Heidelberg (1975). https:\/\/doi.org\/10.1007\/BFb0120696","DOI":"10.1007\/BFb0120696"},{"key":"694_CR10","doi-asserted-by":"publisher","unstructured":"Ben-Tal, A., Teboulle, M.: A smoothing technique for nondifferentiable optimization problems. In: Dolecki, S. (ed.) Optimization. Lecture Notes in Mathematics, vol. 1405, pp. 1\u201311. Springer, Berlin, Heidelberg (1989). https:\/\/doi.org\/10.1007\/BFb0083582","DOI":"10.1007\/BFb0083582"},{"issue":"2","key":"694_CR11","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/s10589-007-9011-6","volume":"37","author":"M Herty","year":"2007","unstructured":"Herty, M., Klar, A., Singh, A.K., Spellucci, P.: Smoothed penalty algorithms for optimization of nonlinear models. Comput. Optim. Appl. 37(2), 157\u2013176 (2007). https:\/\/doi.org\/10.1007\/s10589-007-9011-6","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"694_CR12","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1137\/0804027","volume":"4","author":"M\u00c7 Pinar","year":"1994","unstructured":"Pinar, M.\u00c7., Zenios, S.A.: On smoothing exact penalty functions for convex constrained optimization. SIAM J. Optim. 4(3), 486\u2013511 (1994). https:\/\/doi.org\/10.1137\/0804027","journal-title":"SIAM J. Optim."},{"key":"694_CR13","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s10107-002-0332-z","volume":"96","author":"CC Gonzaga","year":"2003","unstructured":"Gonzaga, C.C., Castillo, R.: A nonlinear programming algorithm based on non-coercive penalty functions. Math. Program. 96, 87\u2013101 (2003). https:\/\/doi.org\/10.1007\/s10107-002-0332-z","journal-title":"Math. Program."},{"issue":"6","key":"694_CR14","doi-asserted-by":"publisher","first-page":"3113","DOI":"10.1016\/j.amc.2012.09.042","volume":"219","author":"S Lian","year":"2012","unstructured":"Lian, S.: Smoothing approximation to $$\\ell _1$$ exact penalty function for inequality constrained optimization. Appl. Math. Comput. 219(6), 3113\u20133121 (2012). https:\/\/doi.org\/10.1016\/j.amc.2012.09.042","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"694_CR15","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s12190-008-0171-z","volume":"30","author":"B Liu","year":"2009","unstructured":"Liu, B.: On smoothing exact penalty functions for nonlinear constrained optimization problems. J. Appl. Math. Comput. 30(1), 259\u2013270 (2009). https:\/\/doi.org\/10.1007\/s12190-008-0171-z","journal-title":"J. Appl. Math. Comput."},{"issue":"7","key":"694_CR16","doi-asserted-by":"publisher","first-page":"806","DOI":"10.1080\/01630563.2011.577262","volume":"32","author":"Z Meng","year":"2011","unstructured":"Meng, Z., Dang, C., Jiang, M., Shen, R.: A smoothing objective penalty function algorithm for inequality constrained optimization problems. Numer. Funct. Anal. Optim. 32(7), 806\u2013820 (2011). https:\/\/doi.org\/10.1080\/01630563.2011.577262","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"1","key":"694_CR17","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF01592244","volume":"71","author":"C Chen","year":"1995","unstructured":"Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71(1), 51\u201369 (1995). https:\/\/doi.org\/10.1007\/BF01592244","journal-title":"Math. Program."},{"issue":"2","key":"694_CR18","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). https:\/\/doi.org\/10.1007\/BF00249052","journal-title":"Comput. Optim. Appl."},{"key":"694_CR19","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/BF01581628","volume":"19","author":"I Zang","year":"1980","unstructured":"Zang, I.: A smoothing-out technique for min-max optimization. Math. Program. 19, 61\u201377 (1980). https:\/\/doi.org\/10.1007\/BF01581628","journal-title":"Math. Program."},{"issue":"13","key":"694_CR20","doi-asserted-by":"publisher","first-page":"3981","DOI":"10.1080\/02331934.2021.1934680","volume":"71","author":"RS Burachik","year":"2021","unstructured":"Burachik, R.S., Kaya, C.Y., Price, C.J.: A primal-dual penalty method via rounded weighted-$$\\ell _1$$ Lagrangian duality. Optimization 71(13), 3981\u20134017 (2021). https:\/\/doi.org\/10.1080\/02331934.2021.1934680","journal-title":"Optimization"},{"key":"694_CR21","doi-asserted-by":"publisher","unstructured":"Lian, S., Duan, Y.: Smoothing of the lower-order exact penalty function for inequality constrained optimization. J. Inequal. Appl. 185 (2016) https:\/\/doi.org\/10.1186\/s13660-016-1126-9","DOI":"10.1186\/s13660-016-1126-9"},{"key":"694_CR22","doi-asserted-by":"publisher","unstructured":"Sahiner, A., Kapusuz, G., Yilmaz, N.: A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numer. Algebra, Control Optim. 6(2), 161\u2013173 (2016) https:\/\/doi.org\/10.3934\/naco.2016006","DOI":"10.3934\/naco.2016006"},{"issue":"1","key":"694_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/01630563.2018.1483948","volume":"40","author":"X Xu","year":"2019","unstructured":"Xu, X., Dang, C., Chan, F.T.S., Wang, Y.: On smoothing $$\\ell _1$$ exact penalty function for constrained optimization problems. Numer. Funct. Anal. Optim. 40(1), 1\u201318 (2019). https:\/\/doi.org\/10.1080\/01630563.2018.1483948","journal-title":"Numer. Funct. Anal. Optim."},{"key":"694_CR24","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s10589-012-9504-9","volume":"55","author":"X Xu","year":"2013","unstructured":"Xu, X., Meng, Z., Sun, J., Huang, L., Shen, R.: A second-order smooth penalty function algorithm for constrained optimization problems. Comput. Optim. Appl. 55, 155\u2013172 (2013). https:\/\/doi.org\/10.1007\/s10589-012-9504-9","journal-title":"Comput. Optim. Appl."},{"key":"694_CR25","doi-asserted-by":"publisher","unstructured":"Solodov, M.V.: Constraint Qualifications. In: Cochran, J.J., Cox Jr, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science. John Wiley & Sons, Hoboken, NJ, USA (2010). https:\/\/doi.org\/10.1002\/9780470400531.eorms0978","DOI":"10.1002\/9780470400531.eorms0978"},{"key":"694_CR26","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/PL00011406","volume":"89","author":"AF Izmailov","year":"2001","unstructured":"Izmailov, A.F., Solodov, M.V.: Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications. Math. Program. 89, 413\u2013435 (2001). https:\/\/doi.org\/10.1007\/PL00011406","journal-title":"Math. Program."},{"key":"694_CR27","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1007\/s10589-016-9845-x","volume":"65","author":"EW Karas","year":"2016","unstructured":"Karas, E.W., Santos, S.A., Svaiter, B.F.: Algebraic rules for computing the regularization parameter of the Levenberg-Marquardt method. Comput. Optim. Appl. 65, 723\u2013751 (2016). https:\/\/doi.org\/10.1007\/s10589-016-9845-x","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"694_CR28","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1137\/110843939","volume":"22","author":"R Andreani","year":"2012","unstructured":"Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: Two new weak constraint qualifications and applications. SIAM J. Optim. 22(3), 1109\u20131135 (2012). https:\/\/doi.org\/10.1137\/110843939","journal-title":"SIAM J. Optim."},{"key":"694_CR29","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s10107-011-0456-0","volume":"135","author":"R Andreani","year":"2012","unstructured":"Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135, 255\u2013273 (2012). https:\/\/doi.org\/10.1007\/s10107-011-0456-0","journal-title":"Math. Program."},{"issue":"5","key":"694_CR30","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1080\/02331934.2010.527972","volume":"61","author":"S Lu","year":"2012","unstructured":"Lu, S.: Relation between the constant rank and the relaxed constant rank constraint qualifications. Optimization 61(5), 555\u2013566 (2012). https:\/\/doi.org\/10.1080\/02331934.2010.527972","journal-title":"Optimization"},{"issue":"4","key":"694_CR31","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1080\/02331930902971377","volume":"60","author":"L Minchenko","year":"2011","unstructured":"Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(4), 429\u2013440 (2011). https:\/\/doi.org\/10.1080\/02331930902971377","journal-title":"Optimization"},{"issue":"4","key":"694_CR32","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1137\/0713043","volume":"13","author":"SM Robinson","year":"1976","unstructured":"Robinson, S.M.: Stability theory for systems of inequalities, part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497\u2013513 (1976). https:\/\/doi.org\/10.1137\/0713043","journal-title":"SIAM J. Numer. Anal."},{"key":"694_CR33","doi-asserted-by":"publisher","unstructured":"Siqueira, A.S., Orban, D.: CUTEst.jl: Julia\u2019s CUTEst interface. Zenodo 1.0.0 (2024) https:\/\/doi.org\/10.5281\/zenodo.1188851","DOI":"10.5281\/zenodo.1188851"},{"key":"694_CR34","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10589-005-1066-7","volume":"31","author":"EG Birgin","year":"2005","unstructured":"Birgin, E.G., Castillo, R.A., Mart\u00ednez, J.M.: Numerical comparison of Augmented Lagrangian algorithms for nonconvex problems. Comput. Optim. Appl. 31, 31\u201355 (2005). https:\/\/doi.org\/10.1007\/s10589-005-1066-7","journal-title":"Comput. Optim. Appl."},{"issue":"24","key":"694_CR35","doi-asserted-by":"publisher","first-page":"615","DOI":"10.21105\/joss.00615","volume":"3","author":"PK Mogensen","year":"2018","unstructured":"Mogensen, P.K., Riseth, A.N.: Optim: A mathematical optimization package for Julia. J. Open Source Softw. 3(24), 615 (2018). https:\/\/doi.org\/10.21105\/joss.00615","journal-title":"J. Open Source Softw."},{"issue":"2","key":"694_CR36","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1007\/s10589-012-9466-y","volume":"53","author":"EG Birgin","year":"2012","unstructured":"Birgin, E.G., Gentil, J.M.: Evaluating bound-constrained minimization software. Comput. Optim. Appl. 53(2), 347\u2013373 (2012). https:\/\/doi.org\/10.1007\/s10589-012-9466-y","journal-title":"Comput. Optim. Appl."},{"key":"694_CR37","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201\u2013213 (2002). https:\/\/doi.org\/10.1007\/s101070100263","journal-title":"Math. Program."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00694-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00694-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00694-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T07:15:51Z","timestamp":1755760551000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00694-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,26]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["694"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00694-9","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2025,5,26]]},"assertion":[{"value":"7 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 May 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}]}}