{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T18:57:52Z","timestamp":1775761072197,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T00:00:00Z","timestamp":1525392000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T00:00:00Z","timestamp":1525392000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000083","name":"Directorate for Computer and Information Science and Engineering","doi-asserted-by":"publisher","award":["1632971"],"award-info":[{"award-number":["1632971"]}],"id":[{"id":"10.13039\/100000083","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2019,7]]},"DOI":"10.1007\/s10107-018-1283-3","type":"journal-article","created":{"date-parts":[[2018,5,4]],"date-time":"2018-05-04T09:18:06Z","timestamp":1525425486000},"page":"95-135","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Structural properties of affine sparsity constraints"],"prefix":"10.1007","volume":"176","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1995-4608","authenticated-orcid":false,"given":"Hongbo","family":"Dong","sequence":"first","affiliation":[]},{"given":"Miju","family":"Ahn","sequence":"additional","affiliation":[]},{"given":"Jong-Shi","family":"Pang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,5,4]]},"reference":[{"key":"1283_CR1","doi-asserted-by":"crossref","unstructured":"Ahn, M., Pang, J.S., Xin, J.: Difference-of-convex learning: directional stationarity, optimality, and sparsity. SIAM J. Optim. Revision under review (as of February 2017)","DOI":"10.1137\/16M1084754"},{"issue":"4","key":"1283_CR2","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1214\/12-STS394","volume":"27","author":"F Bach","year":"2012","unstructured":"Bach, F., Jenatton, R., Mairal, J., Obozinski, G.: Structured sparsity through convex optimization. Stat. Sci. 27(4), 450\u2013468 (2012)","journal-title":"Stat. Sci."},{"key":"1283_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492913000032","volume":"22","author":"P Belotti","year":"2013","unstructured":"Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1\u2013131 (2013)","journal-title":"Acta Numer."},{"issue":"2","key":"1283_CR4","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1214\/15-AOS1388","volume":"44","author":"D Bertsimas","year":"2016","unstructured":"Bertsimas, D., King, A., Mazumder, R.: Best subset selection via a modern optimization lens. Ann. Stat. 44(2), 813\u2013852 (2016)","journal-title":"Ann. Stat."},{"issue":"1","key":"1283_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10589-007-9126-9","volume":"43","author":"D Bertsimas","year":"2009","unstructured":"Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43(1), 1\u201322 (2009)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"1283_CR6","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1214\/13-AOS1096","volume":"43","author":"J Bien","year":"2013","unstructured":"Bien, J., Taylor, J., Tibshirani, R.: A lasso for hierarchical interactions. Ann. Stat. 43(3), 1111\u20131141 (2013)","journal-title":"Ann. Stat."},{"issue":"2","key":"1283_CR7","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/BF02592208","volume":"74","author":"D Bienstock","year":"1996","unstructured":"Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. Ser. A 74(2), 121\u2013140 (1996)","journal-title":"Math. Program. Ser. A"},{"issue":"30","key":"1283_CR8","doi-asserted-by":"publisher","first-page":"12267","DOI":"10.1073\/pnas.0904287106","volume":"106","author":"J Brodie","year":"2009","unstructured":"Brodie, J., Daubechies, I., De Mol, C., Giannone, D., Loris, I.: Sparse and stable Markowitz portfolios. Proc. Natl. Acad. Sci. 106(30), 12267\u201312272 (2009)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"1","key":"1283_CR9","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1137\/140978077","volume":"26","author":"OP Burdakov","year":"2016","unstructured":"Burdakov, O.P., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. 26(1), 397\u2013425 (2016)","journal-title":"SIAM J. Optim."},{"key":"1283_CR10","unstructured":"Chen, C., Li, X., Tolman, C., Wang, S., Ye, Y.: Sparse portfolio selection via quasi-norm regularization. arXiv:1312.6350v1 (2013)"},{"issue":"5","key":"1283_CR11","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1145\/210118.210137","volume":"42","author":"M Conforti","year":"1995","unstructured":"Conforti, M., Cornuejols, G.: A class of logic problems solvable by linear programming. J. ACM 42(5), 1107\u20131112 (1995)","journal-title":"J. ACM"},{"key":"1283_CR12","first-page":"277","volume-title":"Discrete Optimization. Handbooks in Operations Research and Management Science","author":"M Conforti","year":"2005","unstructured":"Conforti, M., Cornuejols, G.: Balanced matrices. In: Aardal, K., Nemhauser, G.L., Weismantel, R. (eds.) Discrete Optimization. Handbooks in Operations Research and Management Science, vol. 12, pp. 277\u2013320. Elsevier, Amsterdam (2005)"},{"issue":"1","key":"1283_CR13","first-page":"55","volume":"30","author":"A d\u2019Aspremont","year":"2008","unstructured":"d\u2019Aspremont, A., Banerjee, O., El Ghaoui, L.: First-order methods for sparse covariance selection. SIAM J. Matrix Anal. Appl. 30(1), 55\u201366 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"1283_CR14","first-page":"587","volume":"16","author":"A-V de Miguel","year":"2006","unstructured":"de Miguel, A.-V., Friedlander, M., Nogales, F.J., Scholtes, S.: A two-sided relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 16(2), 587\u2013609 (2006)","journal-title":"SIAM J. Optim."},{"issue":"456","key":"1283_CR15","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. Am. Stat. Assoc. 96(456), 1348\u20131360 (2001)","journal-title":"J. Am. Stat. Assoc."},{"key":"1283_CR16","unstructured":"Feng, M., Mitchell, J.E., Pang, J.S., Waechter, A., Shen, X.: Complementarity formulations of $$\\ell _0$$-norm optimization problems. Pac. J. Optim. Accepted Aug 2016"},{"issue":"3","key":"1283_CR17","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1093\/biostatistics\/kxm045","volume":"9","author":"JH Friedman","year":"2013","unstructured":"Friedman, J.H., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical Lasso. Biostatistics 9(3), 432\u2013441 (2013)","journal-title":"Biostatistics"},{"key":"1283_CR18","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1080\/00224065.1992.11979383","volume":"24","author":"M Hamada","year":"1992","unstructured":"Hamada, M., Wu, C.F.J.: Analysis of designed experiments with complex aliasing. J. Qual. Technol. 24, 130\u2013137 (1992)","journal-title":"J. Qual. Technol."},{"key":"1283_CR19","doi-asserted-by":"publisher","DOI":"10.1201\/b18401","volume-title":"Statistical Learning with Sparsity: The Lasso and Generalizations","author":"T Hastie","year":"2015","unstructured":"Hastie, T., Tibshirani, R., Wainwright, M.: Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press Taylor & Francis Group, Boca Raton (2015)"},{"issue":"4","key":"1283_CR20","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1214\/12-STS392","volume":"27","author":"J Huang","year":"2012","unstructured":"Huang, J., Breheny, P., Ma, S.: A selective review of group selection in high-dimensional models. Stat. Sci. 27(4), 481\u2013499 (2012)","journal-title":"Stat. Sci."},{"key":"1283_CR21","doi-asserted-by":"crossref","unstructured":"Jacob, L., Obozinski, G., Vert, J.P.: Group lasso with overlap and graph lasso. In: Proceeding of the 26th Annual International Conference on Machine Learning, Montreal, Canada (ICML \u201909, ACM New York) pp. 433\u2013440 (2009)","DOI":"10.1145\/1553374.1553431"},{"issue":"2","key":"1283_CR22","doi-asserted-by":"publisher","first-page":"770","DOI":"10.1137\/100802487","volume":"23","author":"C Kanzow","year":"2013","unstructured":"Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. SIAM J. Optim. 23(2), 770\u2013798 (2013)","journal-title":"SIAM J. Optim."},{"key":"1283_CR23","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.ejor.2014.11.031","volume":"244","author":"HA Le Thi","year":"2015","unstructured":"Le Thi, H.A., Pham, D.T., Vo, X.T.: DC approximation approaches for sparse optimization. Eur. J. Oper. Res. 244, 26\u201346 (2015)","journal-title":"Eur. J. Oper. Res."},{"key":"1283_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3244-0","volume-title":"Generalized Linear Models","author":"P McCullagh","year":"1983","unstructured":"McCullagh, P., Nelder, J.A.: Generalized Linear Models. Chapman & Hall, London (1983)"},{"key":"1283_CR25","volume-title":"Integer and Combinatorial Optimization","author":"G Nemhauser","year":"1999","unstructured":"Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, New York (1999)"},{"key":"1283_CR26","doi-asserted-by":"publisher","unstructured":"Pang, J.S., Razaviyayn, M., Alvarado, A.: Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. https:\/\/doi.org\/10.1287\/moor.2016.0795","DOI":"10.1287\/moor.2016.0795"},{"issue":"2","key":"1283_CR27","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1089\/cmb.2014.0197","volume":"22","author":"H Park","year":"2015","unstructured":"Park, H., Niida, A., Miyano, S., Imoto, S.: Sparse overlapping group Lasso for integrative multi-Omics analysis. J. Comput. Biol. 22(2), 73\u201384 (2015)","journal-title":"J. Comput. Biol."},{"issue":"5","key":"1283_CR28","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1080\/10556780410001709439","volume":"19","author":"D Ralph","year":"2004","unstructured":"Ralph, D., Wright, S.J.: Some properties of regularization and penalization schemes for MPECs. Optim. Methods Softw. 19(5), 527\u2013556 (2004)","journal-title":"Optim. Methods Softw."},{"key":"1283_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"issue":"4","key":"1283_CR30","doi-asserted-by":"publisher","first-page":"918","DOI":"10.1137\/S1052623499361233","volume":"11","author":"S Scholtes","year":"2001","unstructured":"Scholtes, S.: Convergence properties of a regularisation scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918\u2013936 (2001)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1283_CR31","doi-asserted-by":"crossref","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. Stat. Soc. Ser. B (Methodol.) 58(1), 267\u2013288 (1996)","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"issue":"1","key":"1283_CR32","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1111\/j.1467-9868.2005.00490.x","volume":"67","author":"R Tibshirani","year":"2005","unstructured":"Tibshirani, R., Saunders, M.A., Rosset, S., Zhu, J., Knight, K.: Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67(1), 91\u2013108 (2005)","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"1283_CR33","unstructured":"Wang, J., Ye, J.: Multi-layer feature reduction for tree structured group lasso via hierarchical projection. In: Proceedings of the 28th International Conference on Neural Information Processing Systems, Montreal, Canada pp. 1279\u20131287 (2015)"},{"issue":"1","key":"1283_CR34","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","volume":"68","author":"M Yuan","year":"2006","unstructured":"Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B Stat. Methods 68(1), 49\u201367 (2006)","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methods"},{"issue":"2","key":"1283_CR35","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. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"key":"1283_CR36","first-page":"1081","volume":"11","author":"T Zhang","year":"2010","unstructured":"Zhang, T.: Analysis of multi-stage convex relaxation for sparse regularization. J. Mach. Learn. Res. 11, 1081\u20131107 (2010)","journal-title":"J. Mach. Learn. Res."},{"issue":"6A","key":"1283_CR37","doi-asserted-by":"publisher","first-page":"3468","DOI":"10.1214\/07-AOS584","volume":"37","author":"P Zhao","year":"2009","unstructured":"Zhao, P., Rocha, G., Yu, B.: The composite absolute penalties family for grouped and hierarchical variable selection. Ann. Stat. 37(6A), 3468\u20133497 (2009)","journal-title":"Ann. Stat."},{"issue":"1\u20132","key":"1283_CR38","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/s10589-013-9582-3","volume":"59","author":"X Zheng","year":"2014","unstructured":"Zheng, X., Sun, X., Li, D., Sun, J.: Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach. Comput. Optim. Appl. 59(1\u20132), 379\u2013397 (2014)","journal-title":"Comput. Optim. Appl."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-018-1283-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-018-1283-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-018-1283-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,6]],"date-time":"2024-07-06T14:21:09Z","timestamp":1720275669000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-018-1283-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,4]]},"references-count":38,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["1283"],"URL":"https:\/\/doi.org\/10.1007\/s10107-018-1283-3","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,4]]},"assertion":[{"value":"12 May 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 April 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}