{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T14:27:28Z","timestamp":1747664848151,"version":"3.37.3"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T00:00:00Z","timestamp":1683072000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T00:00:00Z","timestamp":1683072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"crossref","award":["12101262"],"award-info":[{"award-number":["12101262"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100021171","name":"Guangdong Basic and Applied Basic Research Foundation","doi-asserted-by":"crossref","award":["2022A1515010263"],"award-info":[{"award-number":["2022A1515010263"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,8]]},"DOI":"10.1007\/s12190-023-01866-8","type":"journal-article","created":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T12:02:15Z","timestamp":1683115335000},"page":"2987-3003","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A modulus-based matrix splitting method for the vertical nonlinear complementarity problem"],"prefix":"10.1007","volume":"69","author":[{"given":"Shuilian","family":"Xie","sequence":"first","affiliation":[]},{"given":"Zhen-Ping","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Hongru","family":"Xu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,3]]},"reference":[{"key":"1866_CR1","first-page":"571","volume":"19","author":"M Sun","year":"1989","unstructured":"Sun, M.: Monotonicity of Mangasarian\u2019s iterative algorithm for generalized linear complementarity problems. J. Math. Anal. Appl. 19, 571\u2013584 (1989)","journal-title":"J. Math. Anal. Appl."},{"key":"1866_CR2","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":"1866_CR3","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF02061065","volume":"44","author":"AA Ebiefung","year":"1993","unstructured":"Ebiefung, A.A., Kostreva, M.M.: The generalized Lenotief input-ouput model and its application to the choice of the new technology. Ann. Oper. Res. 44, 161\u2013172 (1993)","journal-title":"Ann. Oper. Res."},{"key":"1866_CR4","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. Dyn. Control. 32, 1754\u20131779 (2008)","journal-title":"J. Econom. Dyn. Control."},{"key":"1866_CR5","first-page":"45","volume":"21","author":"KP Oh","year":"1986","unstructured":"Oh, K.P.: The formulation of the mixed lubrication problem as a generalized nonlinear complementarity problem. J. Tribol. 21, 45\u201366 (1986)","journal-title":"J. Tribol."},{"key":"1866_CR6","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1137\/S0895479897329837","volume":"21","author":"HD 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, 45\u201366 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1866_CR7","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s101070050104","volume":"86","author":"JM Peng","year":"1999","unstructured":"Peng, J.M., Lin, Z.: A non-interior continuous method for generalized linear complementarity problems. Math. Program. Ser. A 86, 533\u2013563 (1999)","journal-title":"Math. Program. Ser. A"},{"key":"1866_CR8","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1007\/s10957-021-01922-y","volume":"193","author":"F Mezzadri","year":"2022","unstructured":"Mezzadri, F., Galligani, E.: Projected splitting methods for vertical linear complementarity problems. J. Optim. Theory Appl. 193, 598\u2013620 (2022)","journal-title":"J. Optim. Theory Appl."},{"key":"1866_CR9","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1002\/nla.680","volume":"17","author":"ZZ 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":"1866_CR10","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1002\/nla.1835","volume":"20","author":"ZZ 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":"1866_CR11","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2105","volume":"24","author":"ZZ Bai","year":"2017","unstructured":"Bai, Z.Z., Zhang, L.L.: Modulus-based multigrid methods for linear complementarity problems. Numer. Linear Algebra Appl. 24, e2105 (2017)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1866_CR12","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.F.: Accelerated modulus-based matrix iteration methods for linear complementarity problem. Numer. Algorithms 64, 245\u2013262 (2013)","journal-title":"Numer. Algorithms"},{"key":"1866_CR13","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s11075-010-9416-7","volume":"57","author":"LL Zhang","year":"2011","unstructured":"Zhang, L.L.: Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer. Algorithms 57, 83\u201399 (2011)","journal-title":"Numer. Algorithms"},{"key":"1866_CR14","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1080\/03081087.2015.1087457","volume":"64","author":"W Li","year":"2016","unstructured":"Li, W., Zheng, H.: A preconditioned modulus-based iteration method for solving linear complementarity problems of H-matrices. Linear Multilinear Algebra 64, 1390\u20131403 (2016)","journal-title":"Linear Multilinear Algebra"},{"key":"1866_CR15","doi-asserted-by":"publisher","first-page":"1547","DOI":"10.1007\/s11075-021-01240-4","volume":"90","author":"F Mezzadri","year":"2022","unstructured":"Mezzadri, F.: A modulus-based formulation for the vertical linear complementarity problem. Numer. Algorithms 90, 1547\u20131568 (2022)","journal-title":"Numer. Algorithms"},{"key":"1866_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108344","volume":"134","author":"J He","year":"2022","unstructured":"He, J., Vong, S.W.: A new kind of modulus-based matrix splitting methods for vertical linear complementarity problems. Appl. Math. Lett. 134, 108344 (2022)","journal-title":"Appl. Math. Lett."},{"key":"1866_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-022-01436-2","author":"H Zheng","year":"2022","unstructured":"Zheng, H., Zhang, Y., Lu, X., Vong, S.: Modulus-based synchronous multi-splitting iteration methods for large sparse vertical linear complementarity problems. Numer. Algorithms (2022). https:\/\/doi.org\/10.1007\/s11075-022-01436-2","journal-title":"Numer. Algorithms"},{"key":"1866_CR18","first-page":"14","volume":"2022","author":"Y Song","year":"1882","unstructured":"Song, Y., Zheng, H., Lu, X., Vong, S.: A two-step iteration method for vertical linear complementarity problems. Symmetry 2022, 14 (1882)","journal-title":"Symmetry"},{"key":"1866_CR19","doi-asserted-by":"publisher","first-page":"108444","DOI":"10.1016\/j.aml.2022.108444","volume":"135","author":"WX Guo","year":"2023","unstructured":"Guo, W.X., Zheng, H., Peng, X.F.: New convergence results of the modulus-based methods for vertical linear complementarity problems. Appl. Math. Lett. 135, 108444 (2023)","journal-title":"Appl. Math. Lett."},{"key":"1866_CR20","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1006\/jmaa.1998.6205","volume":"230","author":"HD Qi","year":"1999","unstructured":"Qi, H.D., Liao, L.Z., Lin, Z.H.: Regularized smoothing approximations to vertical nonlinear complementarity problems. J. Math. Anal. Appl. 230, 261\u2013276 (1999)","journal-title":"J. Math. Anal. Appl."},{"key":"1866_CR21","doi-asserted-by":"publisher","first-page":"6650","DOI":"10.3934\/math.2022371","volume":"7","author":"B Hou","year":"2022","unstructured":"Hou, B., Zhang, J., Qiu, C.: A neural network for a generalized vertical complementarity problem. AIMS Math. 7, 6650\u20136668 (2022)","journal-title":"AIMS Math."},{"key":"1866_CR22","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"J Ortega","year":"1970","unstructured":"Ortega, J., Rheinboldt, W.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01866-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01866-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01866-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,19]],"date-time":"2023-07-19T19:23:45Z","timestamp":1689794625000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01866-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,3]]},"references-count":22,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["1866"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01866-8","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2023,5,3]]},"assertion":[{"value":"15 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 March 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}]}}