{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:49:20Z","timestamp":1767340160582,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["RTI2018-097580-B-I00."],"award-info":[{"award-number":["RTI2018-097580-B-I00."]}],"id":[{"id":"10.13039\/501100004837","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":[[2024,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Multiple variable splitting is a general technique for decomposing problems by using copies of variables and additional linking constraints that equate their values. The resulting large optimization problem can be solved with a specialized interior-point method that exploits the problem structure and computes the Newton direction with a combination of direct and iterative solvers (i.e. Cholesky factorizations and preconditioned conjugate gradients for linear systems related to, respectively, subproblems and new linking constraints). The present work applies this method to solving real-world binary classification and novelty (or outlier) detection problems by means of, respectively, two-class and one-class linear support vector machines (SVMs). Unlike previous interior-point approaches for SVMs, which were practical only with low-dimensional points, the new proposal can also deal with high-dimensional data. The new method is compared with state-of-the-art solvers for SVMs that are based on either interior-point algorithms (such as SVM-OOPS) or specific algorithms developed by the machine learning community (such as LIBSVM and LIBLINEAR). The computational results show that, for two-class SVMs, the new proposal is competitive not only against previous interior-point methods\u2014and much more efficient than they are with high-dimensional data\u2014but also against LIBSVM, whereas LIBLINEAR generally outperformed the proposal. For one-class SVMs, the new method consistently outperformed all other approaches, in terms of either solution time or solution quality.<\/jats:p>","DOI":"10.1007\/s10957-022-02103-1","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T17:04:01Z","timestamp":1664471041000},"page":"237-270","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["New Interior-Point Approach for One- and Two-Class Linear Support Vector Machines Using Multiple Variable Splitting"],"prefix":"10.1007","volume":"202","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3573-4568","authenticated-orcid":false,"given":"Jordi","family":"Castro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,29]]},"reference":[{"key":"2103_CR1","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1007\/s10957-013-0458-6","volume":"164","author":"A Astorino","year":"2015","unstructured":"Astorino, A., Fuduli, A.: Support vector machine polyhedral separability in semisupervised learning. J. Optim. Theory Appl. 164, 1039\u20131050 (2015). https:\/\/doi.org\/10.1007\/s10957-013-0458-6","journal-title":"J. Optim. Theory Appl."},{"key":"2103_CR2","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1137\/16M1080173","volume":"60","author":"L Bottou","year":"2018","unstructured":"Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. SIAM Rev. 60, 223\u2013311 (2018). https:\/\/doi.org\/10.1137\/16M1080173","journal-title":"SIAM Rev."},{"key":"2103_CR3","doi-asserted-by":"publisher","first-page":"852","DOI":"10.1137\/S1052623498341879","volume":"10","author":"J Castro","year":"2000","unstructured":"Castro, J.: A specialized interior-point algorithm for multicommodity network flows. SIAM J. Optim. 10, 852\u2013877 (2000). https:\/\/doi.org\/10.1137\/S1052623498341879","journal-title":"SIAM J. Optim."},{"key":"2103_CR4","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10589-006-9000-1","volume":"36","author":"J Castro","year":"2007","unstructured":"Castro, J.: An interior-point approach for primal block-angular problems. Comput. Optim. Appl. 36, 195\u2013219 (2007). https:\/\/doi.org\/10.1007\/s10589-006-9000-1","journal-title":"Comput. Optim. Appl."},{"key":"2103_CR5","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1080\/10556788.2015.1050014","volume":"31","author":"J Castro","year":"2016","unstructured":"Castro, J.: Interior-point solver for convex separable block-angular problems. Optim. Methods Softw. 31, 88\u2013109 (2016). https:\/\/doi.org\/10.1080\/10556788.2015.1050014","journal-title":"Optim. Methods Softw."},{"key":"2103_CR6","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/s10107-010-0341-2","volume":"130","author":"J Castro","year":"2011","unstructured":"Castro, J., Cuesta, J.: Quadratic regularizations in an interior-point method for primal block-angular problems. Math. Program. 130, 415\u2013445 (2011). https:\/\/doi.org\/10.1007\/s10107-010-0341-2","journal-title":"Math. Program."},{"key":"2103_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1961189.1961199","volume":"2","author":"C-C Chang","year":"2011","unstructured":"Chang, C.-C., Lin, C.-J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. 2, 1\u201327 (2011). https:\/\/doi.org\/10.1145\/1961189.1961199","journal-title":"ACM Trans. Intell. Syst. Technol."},{"key":"2103_CR8","doi-asserted-by":"publisher","unstructured":"Chou, H.-Y., Lin, P.-Y., Lin, C.-J.: Dual coordinate-descent methods for linear one-class SVM and SVDD. In: Proceedings of the 2020 SIAM International Conference on Data Mining, pp. 181\u2013189 (2020). https:\/\/doi.org\/10.1137\/1.9781611976236.21","DOI":"10.1137\/1.9781611976236.21"},{"key":"2103_CR9","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/BF00994018","volume":"20","author":"C Cortes","year":"1995","unstructured":"Cortes, C., Vapnik, V.: Support vector networks. Mach. Learn. 20, 273\u2013297 (1995). https:\/\/doi.org\/10.1007\/BF00994018","journal-title":"Mach. Learn."},{"key":"2103_CR10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801389","volume-title":"An Introduction to Support Vector Machines and Other Kernel-based Learning Methods","author":"N Cristianini","year":"2000","unstructured":"Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge (2000). https:\/\/doi.org\/10.1017\/CBO9780511801389"},{"key":"2103_CR11","first-page":"1871","volume":"9","author":"R-E Fan","year":"2008","unstructured":"Fan, R.-E., Chang, K.-W., Hsieh, C.-J., Wang, X.-R., Lin, C.-J.: LIBLINEAR: a library for large linear classification. J. Mach. Learn. Res. 9, 1871\u20131874 (2008)","journal-title":"J. Mach. Learn. Res."},{"key":"2103_CR12","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1137\/S1052623400374379","volume":"13","author":"M Ferris","year":"2003","unstructured":"Ferris, M., Munson, T.: Interior point methods for massive support vector machines. SIAM J. Optim. 13, 783\u2013804 (2003). https:\/\/doi.org\/10.1137\/S1052623400374379","journal-title":"SIAM J. Optim."},{"key":"2103_CR13","doi-asserted-by":"crossref","unstructured":"Gertz, E.M., Griffin, J.D.: Support vector machine classifiers for large data sets. Argonne National Laboratory, Technical Report ANL\/MCS-TM-289. https:\/\/www.osti.gov\/biblio\/881587 (2005)","DOI":"10.2172\/881587"},{"key":"2103_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-003-0377-7","volume":"99","author":"D Goldfarb","year":"2004","unstructured":"Goldfarb, D., Scheinberg, K.: A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming. Math. Program. 99, 1\u201334 (2004). https:\/\/doi.org\/10.1007\/s10107-003-0377-7","journal-title":"Math. Program."},{"key":"2103_CR15","unstructured":"Goldfarb, D., Scheinberg, K.: Solving structured convex quadratic programs by interior point methods with application to support vector machines and portfolio optimization. IBM Research Report RC23773 (W0511-025) (2005)"},{"key":"2103_CR16","doi-asserted-by":"publisher","first-page":"1510","DOI":"10.1137\/120886017","volume":"23","author":"J Gondzio","year":"2013","unstructured":"Gondzio, J.: Convergence analysis of an inexact feasible interior point method for convex quadratic programming. SIAM J. Optim. 23, 1510\u20131527 (2013). https:\/\/doi.org\/10.1137\/120886017","journal-title":"SIAM J. Optim."},{"key":"2103_CR17","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1080\/10556789808805689","volume":"9","author":"C M\u00e9sz\u00e1ros","year":"1998","unstructured":"M\u00e9sz\u00e1ros, C.: On free variables in interior point methods. Optim. Methods Softw. 9, 121\u2013139 (1998). https:\/\/doi.org\/10.1080\/10556789808805689","journal-title":"Optim. Methods Softw."},{"key":"2103_CR18","series-title":"Frontiers of Computer Science","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-2112-3","volume-title":"Introduction to Parallel and Vector Solutions of Linear Systems","author":"JM Ortega","year":"1988","unstructured":"Ortega, J.M.: Introduction to Parallel and Vector Solutions of Linear Systems. Frontiers of Computer Science, Springer, Boston (1988). https:\/\/doi.org\/10.1007\/978-1-4899-2112-3"},{"key":"2103_CR19","volume-title":"Advances in Kernel Methods: Support Vector Learning, 185\u201320","author":"J Platt","year":"1999","unstructured":"Platt, J.: Fast training of support vector machines using sequential minimal optimization. In: Sch\u00f6lkopf, B., Burges, C.J.C., Smola, A.J. (eds.) Advances in Kernel Methods: Support Vector Learning, 185\u201320. MIT Press, Cambridge (1999)"},{"key":"2103_CR20","doi-asserted-by":"publisher","first-page":"1443","DOI":"10.1162\/089976601750264965","volume":"13","author":"B Sch\u00f6lkopf","year":"2001","unstructured":"Sch\u00f6lkopf, B., Platt, J.C., Shawe-Taylor, J., Smola, A.J.: Estimating the support of a high-dimensional distribution. Neural Comput. 13, 1443\u20131471 (2001). https:\/\/doi.org\/10.1162\/089976601750264965","journal-title":"Neural Comput."},{"key":"2103_CR21","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/s10589-009-9296-8","volume":"49","author":"K Woodsend","year":"2011","unstructured":"Woodsend, K., Gondzio, J.: Exploiting separability in large-scale linear support vector machine training. Comput. Optim. Appl. 49, 241\u2013269 (2011). https:\/\/doi.org\/10.1007\/s10589-009-9296-8","journal-title":"Comput. Optim. Appl."},{"key":"2103_CR22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971453","volume-title":"Primal-Dual Interior-Point Methods","author":"SJ Wright","year":"1997","unstructured":"Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997). https:\/\/doi.org\/10.1137\/1.9781611971453"},{"key":"2103_CR23","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10107-015-0892-3","volume":"151","author":"SJ Wright","year":"2015","unstructured":"Wright, S.J.: Coordinate descent algorithms. Math. Program. 151, 3\u201334 (2015). https:\/\/doi.org\/10.1007\/s10107-015-0892-3","journal-title":"Math. Program."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02103-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-022-02103-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02103-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,21]],"date-time":"2024-07-21T08:01:47Z","timestamp":1721548907000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-022-02103-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,29]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["2103"],"URL":"https:\/\/doi.org\/10.1007\/s10957-022-02103-1","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2022,9,29]]},"assertion":[{"value":"17 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 September 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}