{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T01:43:00Z","timestamp":1782524580919,"version":"3.54.5"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,1,7]],"date-time":"2022-01-07T00:00:00Z","timestamp":1641513600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,7]],"date-time":"2022-01-07T00:00:00Z","timestamp":1641513600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,8]]},"DOI":"10.1007\/s11075-021-01240-4","type":"journal-article","created":{"date-parts":[[2022,1,7]],"date-time":"2022-01-07T00:04:44Z","timestamp":1641513884000},"page":"1547-1568","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["A modulus-based formulation for the vertical linear complementarity problem"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9779-7647","authenticated-orcid":false,"given":"Francesco","family":"Mezzadri","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,1,7]]},"reference":[{"key":"1240_CR1","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/S0021-9800(70)80010-2","volume":"8","author":"RW Cottle","year":"1970","unstructured":"Cottle, R.W., Dantzig, G.B.: A generalization of the linear complementarity problem. J. Combin. Theory 8, 79\u201390 (1970)","journal-title":"J. Combin. Theory"},{"key":"1240_CR2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719000","volume-title":"The Linear Complementarity Problem, Classics in Applied Mathematics","author":"RW Cottle","year":"2009","unstructured":"Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem, Classics in Applied Mathematics. SIAM, Philadelphia (2009)"},{"key":"1240_CR3","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1137\/S0895479892237859","volume":"15","author":"MS Gowda","year":"1994","unstructured":"Gowda, M.S., Sznajder, R.: The generalized order linear complementarity problem. SIAM J. Matrix Anal. Appl. 15, 779\u2013795 (1994)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1240_CR4","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/0022-247X(89)90347-8","volume":"144","author":"M Sun","year":"1989","unstructured":"Sun, M.: Monotonicity of Mangasarian\u2019s iterative algorithm for generalized linear complementarity problems. J. Math. Anal. Appl. 144, 249\u2013260 (1989)","journal-title":"J. Math. Anal. Appl."},{"key":"1240_CR5","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1137\/0122030","volume":"22","author":"T Fujisawa","year":"1972","unstructured":"Fujisawa, T., Kuh, E.S.: Piecewise-linear theory of nonlinear networks. SIAM J. Appl. Math. 22, 307\u2013328 (1972)","journal-title":"SIAM J. Appl. Math."},{"key":"1240_CR6","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1109\/TCT.1972.1083550","volume":"19","author":"T Fujisawa","year":"1972","unstructured":"Fujisawa, T., Kuh, E.S., Ohtsuki, T.: A sparse matrix method for analysis of piecewise-linear resistive networks. IEEE Trans. Circ. Theory 19, 571\u2013584 (1972)","journal-title":"IEEE Trans. Circ. Theory"},{"key":"1240_CR7","doi-asserted-by":"publisher","first-page":"1754","DOI":"10.1016\/j.jedc.2007.04.010","volume":"32","author":"T Nagae","year":"2008","unstructured":"Nagae, T., Akamatsu, T.: A generalized complementarity approach to solving real option problems. J. Econom. Dynam. Control 32, 1754\u20131779 (2008)","journal-title":"J. Econom. Dynam. Control"},{"key":"1240_CR8","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1115\/1.3261274","volume":"108","author":"KP Oh","year":"1986","unstructured":"Oh, K.P.: The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem. J. Tribol. 108, 598\u2013604 (1986)","journal-title":"J. Tribol."},{"issue":"1","key":"1240_CR9","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1137\/S0895479897329837","volume":"21","author":"H-D Qi","year":"1999","unstructured":"Qi, H.-D., Liao, L.-Z.: A smoothing Newton method for extended vertical linear complementarity problems. SIAM J. Matrix Anal. Appl. 21(1), 45\u201366 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"6","key":"1240_CR10","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1007\/BF02437876","volume":"24","author":"Z-L Ping","year":"2003","unstructured":"Ping, Z.-L., You, G.-Z.: Global linear and quadratic one-step smoothing Newton method for vertical linear complementarity problems. Appl. Math. Mech. (English Ed.) 24(6), 738\u2013746 (2003)","journal-title":"Appl. Math. Mech. (English Ed.)"},{"key":"1240_CR11","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s101070050104","volume":"86","author":"J-M Peng","year":"1999","unstructured":"Peng, J.-M., Lin, Z.: A non-interior continuation method for generalized linear complementarity problems. Math. Program. Ser. A 86, 533\u2013563 (1999)","journal-title":"Math. Program. Ser. A"},{"key":"1240_CR12","doi-asserted-by":"crossref","unstructured":"Mohan, S.R., Neogy, S.K., Sridhar, R.: The generalized linear complementarity problem revisited. Math. Program., 197\u2013218 (1996)","DOI":"10.1007\/BF02592211"},{"key":"1240_CR13","first-page":"255","volume":"69","author":"AA Ebiefung","year":"1995","unstructured":"Ebiefung, A.A.: Nonlinear mappings associated with the generalized linear complementarity problem. Math. Program. 69, 255\u2013268 (1995)","journal-title":"Math. Program."},{"key":"1240_CR14","doi-asserted-by":"publisher","unstructured":"Mezzadri, F., Galligani, E.: Projected splitting methods for vertical linear complementarity problems, J. Optim. Theory Appl. (in press), https:\/\/doi.org\/10.1007\/s10957-021-01922-y (2021)","DOI":"10.1007\/s10957-021-01922-y"},{"issue":"2","key":"1240_CR15","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF00934765","volume":"37","author":"JS Pang","year":"1982","unstructured":"Pang, J.S.: On the convergence of a basic iterative method for the implicit complementarity problem. J. Optim. Theory Appl. 37(2), 149\u2013162 (1982)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"1240_CR16","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1137\/S1052623494275586","volume":"6","author":"SR Mohan","year":"1996","unstructured":"Mohan, S.R., Neogy, S.K.: Algorithms for the generalized linear complementarity problem with a vertical block z-matrix. SIAM J. Optim. 6(4), 994\u20131006 (1996)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1240_CR17","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/S0895479894271147","volume":"18","author":"SR Mohan","year":"1997","unstructured":"Mohan, S.R., Neogy, S.K.: Vertical block hidden Z-matrices and the generalized linear complementarity problem. SIAM J. Matrix Anal. Appl. 18(1), 181\u2013190 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1240_CR18","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1080\/10556789908805739","volume":"10","author":"AA Ebiefung","year":"1999","unstructured":"Ebiefung, A.A.: The vertical linear complementarity problem associated with P0-matrices. Optim. Methods Softw. 10, 747\u2013761 (1999)","journal-title":"Optim. Methods Softw."},{"key":"1240_CR19","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1002\/nla.609","volume":"16","author":"J-L Dong","year":"2009","unstructured":"Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129\u2013143 (2009)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1240_CR20","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1002\/nla.680","volume":"17","author":"Z-Z Bai","year":"2010","unstructured":"Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917\u2013933 (2010)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1240_CR21","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s11075-012-9664-9","volume":"64","author":"N Zheng","year":"2013","unstructured":"Zheng, N., Yin, J.: Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Algoritm. 64, 245\u2013262 (2013)","journal-title":"Numer. Algoritm."},{"issue":"1","key":"1240_CR22","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s11075-016-0142-7","volume":"74","author":"H Zheng","year":"2017","unstructured":"Zheng, H., Li, W., Vong, S.: A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems. Numer. Algoritm. 74(1), 137\u2013152 (2017)","journal-title":"Numer. Algoritm."},{"issue":"1","key":"1240_CR23","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s11075-010-9416-7","volume":"57","author":"L Zhang","year":"2011","unstructured":"Zhang, L.: Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer. Algoritm. 57(1), 83\u201399 (2011)","journal-title":"Numer. Algoritm."},{"key":"1240_CR24","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1002\/nla.1835","volume":"20","author":"Z-Z Bai","year":"2013","unstructured":"Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425\u2013439 (2013)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1240_CR25","first-page":"e2105:1","volume":"24","author":"Z-Z Bai","year":"2017","unstructured":"Bai, Z.-Z., Zhang, L.-L.: Modulus-based multigrid methods for linear complementarity problems. Numer Linear Algebra Appl. 24, e2105:1\u201315 (2017)","journal-title":"Numer Linear Algebra Appl."},{"key":"1240_CR26","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.cam.2015.04.006","volume":"288","author":"H Zheng","year":"2015","unstructured":"Zheng, H., Li, W.: The modulus-based nonsmooth Newton\u2019s method for solving linear complementarity problems. J. Comput. Appl. Math 288, 116\u2013126 (2015)","journal-title":"J. Comput. Appl. Math"},{"key":"1240_CR27","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.laa.2016.01.032","volume":"495","author":"H Zheng","year":"2016","unstructured":"Zheng, H., Li, W., Qu, W.: A non-modulus linear method for solving the linear complementarity problem. Linear Algebra Appl. 495, 38\u201350 (2016)","journal-title":"Linear Algebra Appl."},{"key":"1240_CR28","first-page":"34","volume":"271","author":"Z Xia","year":"2015","unstructured":"Xia, Z., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem. Appl. Math. Comput. 271, 34\u201342 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"1240_CR29","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1002\/nla.2039","volume":"23","author":"N Huang","year":"2016","unstructured":"Huang, N., Ma, C.-F.: The modulus-based matrix splitting algorithms for a class of weakly nonlinear complementarity problems. Numer. Linear Algebra Appl. 23(3), 558\u2013569 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"4","key":"1240_CR30","doi-asserted-by":"publisher","first-page":"1481","DOI":"10.1007\/s10092-017-0236-1","volume":"54","author":"H Zheng","year":"2017","unstructured":"Zheng, H.: Improved convergence theorems of modulus-based matrix splitting iteration method for nonlinear complementarity problems of H-matrices. Calcolo 54(4), 1481\u20131490 (2017)","journal-title":"Calcolo"},{"key":"1240_CR31","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/s11075-016-0243-3","volume":"75","author":"R Li","year":"2017","unstructured":"Li, R., Yin, J.-F.: Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems. Numer. Algoritm. 75, 339\u2013358 (2017)","journal-title":"Numer. Algoritm."},{"issue":"1","key":"1240_CR32","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s11075-019-00677-y","volume":"83","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri, F., Galligani, E.: Modulus-based matrix splitting methods for horizontal linear complementarity problems. Numer. Algoritm. 83(1), 201\u2013219 (2020)","journal-title":"Numer. Algoritm."},{"key":"1240_CR33","first-page":"1","volume":"369","author":"H Zheng","year":"2020","unstructured":"Zheng, H., Vong, S.: On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of H+-matrices. Appl. Math. Comput. 369, 1\u20136 (2020)","journal-title":"Appl. Math. Comput."},{"issue":"e2319","key":"1240_CR34","first-page":"1","volume":"27","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri, F.: Modulus-based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers. Numer. Linear Algebra Appl. 27(e2319), 1\u201315 (2020)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1240_CR35","doi-asserted-by":"publisher","first-page":"1791","DOI":"10.1007\/s11075-020-00954-1","volume":"86","author":"H Zheng","year":"2021","unstructured":"Zheng, H., Vong, S.: A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems. Numer. Algorithm. 86, 1791\u20131810 (2021)","journal-title":"Numer. Algorithm."},{"issue":"2","key":"1240_CR36","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s11075-018-0614-z","volume":"82","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Vong, S.: A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems. Numer. Algorithm. 82 (2), 573\u2013592 (2019)","journal-title":"Numer. Algorithm."},{"key":"1240_CR37","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.apnum.2018.05.017","volume":"132","author":"S-L Wu","year":"2018","unstructured":"Wu, S.-L., Guo, P.: Modulus-based matrix splitting algorithms for the quasi-complementarity problems. Appl. Numer. Math 132, 127\u2013137 (2018)","journal-title":"Appl. Numer. Math"},{"key":"1240_CR38","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"RA Horn","year":"1991","unstructured":"Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)"},{"key":"1240_CR39","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1137\/S0895479897324032","volume":"21","author":"Z-Z Bai","year":"1999","unstructured":"Bai, Z.-Z.: On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J. Matrix Anal. Appl. 21, 67\u201378 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1240_CR40","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF01385865","volume":"63","author":"A Frommer","year":"1992","unstructured":"Frommer, A., Szyld, D.B.: H-splittings and two-stage iterative methods. Numer. Math. 63, 345\u2013356 (1992)","journal-title":"Numer. Math."},{"key":"1240_CR41","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1016\/0024-3795(93)00184-2","volume":"223-224","author":"R Sznajder","year":"1995","unstructured":"Sznajder, R., Gowda, M.S.: Generalizations of P0- and P-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223-224, 695\u2013715 (1995)","journal-title":"Linear Algebra Appl."},{"key":"1240_CR42","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1016\/j.laa.2021.03.016","volume":"621","author":"F Mezzadri","year":"2021","unstructured":"Mezzadri, F., Galligani, E.: A generalization of irreducibility and diagonal dominance with applications to horizontal and vertical linear complementarity problems. Linear Algebra Appl. 621, 214\u2013234 (2021)","journal-title":"Linear Algebra Appl."},{"key":"1240_CR43","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1016\/j.matcom.2020.01.014","volume":"176","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri, F., Galligani, E.: On the convergence of modulus-based matrix splitting methods for horizontal linear complementarity problems in hydrodynamic lubrication. Math. Comput. Simul. 176, 226\u2013242 (2020)","journal-title":"Math. Comput. Simul."},{"key":"1240_CR44","first-page":"126165","volume":"402","author":"H Zheng","year":"2021","unstructured":"Zheng, H., Vong, S.: On the modulus-based successive overrelaxation iteration method for horizontal linear complementarity problems arising from hydrodynamic lubrication. Appl. Math. Comput. 402, 126165 (2021)","journal-title":"Appl. Math. Comput."},{"key":"1240_CR45","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1007\/s42452-021-04238-8","volume":"3","author":"S Miyamoto","year":"2021","unstructured":"Miyamoto, S., Yamashita, M.: An improved convergence based on accelerated modulus-based Gauss\u2013Seidel method for interactive rigid body simulations. SN Appl. Sci. 3, 266 (2021)","journal-title":"SN Appl. Sci."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01240-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01240-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01240-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T08:27:35Z","timestamp":1657873655000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01240-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,7]]},"references-count":45,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["1240"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01240-4","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,7]]},"assertion":[{"value":"22 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}