{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T11:49:19Z","timestamp":1768736959160,"version":"3.49.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T00:00:00Z","timestamp":1622678400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T00:00:00Z","timestamp":1622678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s11075-021-01136-3","type":"journal-article","created":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T17:02:49Z","timestamp":1622739769000},"page":"839-853","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A generalized modulus-based Newton method for solving a class of non-linear complementarity problems with P-matrices"],"prefix":"10.1007","volume":"89","author":[{"given":"Rui","family":"Li","sequence":"first","affiliation":[]},{"given":"Zhi-Lin","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jun-Feng","family":"Yin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,3]]},"reference":[{"key":"1136_CR1","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":"1136_CR2","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s10092-016-0175-2","volume":"54","author":"S-L Wu","year":"2017","unstructured":"Wu, S.-L., Li, C.-X.: A generalized Newton method for non-Hermitian positive definite linear complementarity problem. Calcolo 54, 43\u201356 (2017)","journal-title":"Calcolo"},{"key":"1136_CR3","volume-title":"Iterative solution of nonlinear equations in several variables","author":"JM Ortega","year":"1970","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic, New York (1970)"},{"key":"1136_CR4","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0898-1221(95)00190-5","volume":"31","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: Parallel nonlinear AOR method and its convergence. Comput. Math. Appl. 31, 21\u201331 (1996)","journal-title":"Comput. Math. Appl."},{"key":"1136_CR5","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1137\/S0036144595285963","volume":"39","author":"MC Ferris","year":"1997","unstructured":"Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669\u2013713 (1997)","journal-title":"SIAM Rev."},{"key":"1136_CR6","volume-title":"Finite-dimensional variational inequalities and complementarity problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems. Springer, New York (2003)"},{"key":"1136_CR7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719000","volume-title":"The linear complementarity problem","author":"RW Cottle","year":"2009","unstructured":"Cottle, R.W., Pang, J.-S., Stone, R.E.: The linear complementarity problem. Academic, San Diego (2009)"},{"key":"1136_CR8","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1137\/0309028","volume":"9","author":"CW Cryer","year":"1971","unstructured":"Cryer, C.W.: The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control Optim. 9, 385\u2013392 (1971)","journal-title":"SIAM J. Control Optim."},{"key":"1136_CR9","first-page":"228","volume":"5","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: On the monotone convergence of the projected iteration methods for linear complementarity problems. Numer. Math. J. Chinese Univ. (English Ser.) 5, 228\u2013233 (1996)","journal-title":"Numer. Math. J. Chinese Univ. (English Ser.)"},{"key":"1136_CR10","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/s10957-014-0589-4","volume":"165","author":"A Hadjidimos","year":"2015","unstructured":"Hadjidimos, A., Tzoumas, M.: On the solution of the linear complementarity problem by the generalized accelerated overrelaxation iterative method. J. Optim. Theory Appl. 165, 545\u2013562 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1136_CR11","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0898-1221(96)00169-1","volume":"32","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z., Wang, D.-R.: A class of parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems. Comput. Math. Appl. 32, 79\u201395 (1996)","journal-title":"Comput. Math. Appl."},{"key":"1136_CR12","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1080\/00207169708804569","volume":"63","author":"Z-Z Bai","year":"1997","unstructured":"Bai, Z.-Z., Evans, D.J.: Matrix multisplitting relaxation methods for linear complementarity problems. Int. J. Comput. Math. 63, 309\u2013326 (1997)","journal-title":"Int. J. Comput. Math."},{"key":"1136_CR13","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":"J. Matrix Anal. Appl."},{"key":"1136_CR14","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1017\/S0004972710000389","volume":"28","author":"Y-J Jiang","year":"2010","unstructured":"Jiang, Y.-J., Zeng, J.-P.: A multiplicative Schwarz algorithm for the nonlinear complementarity problem with an M-function. B. Aust. Math. Soc. 28, 353\u2013366 (2010)","journal-title":"B. Aust. Math. Soc."},{"key":"1136_CR15","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1080\/10556780310001636369","volume":"19","author":"C Kanzow","year":"2004","unstructured":"Kanzow, C.: Inexact semismooth Newton methods for large-scale complementarity problems. Optim. Method Softw. 19, 309\u2013325 (2004)","journal-title":"Optim. Method Softw."},{"key":"1136_CR16","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1016\/j.cam.2010.08.012","volume":"235","author":"Z Sun","year":"2011","unstructured":"Sun, Z., Zeng, J.-P.: A monotone semismooth Newton type method for a class of complementarity problems. J. Comput. Appl. Math. 235, 1261\u20131274 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"1136_CR17","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":"1136_CR18","volume-title":"Piecewise-linear modelling and analysis","author":"WMG van Bokhoven","year":"1981","unstructured":"van Bokhoven, W.M.G.: Piecewise-linear modelling and analysis. Proeschrift, Eindhoven (1981)"},{"key":"1136_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":"1136_CR20","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/j.laa.2009.02.024","volume":"431","author":"A Hadjidimos","year":"2009","unstructured":"Hadjidimos, A., Tzoumas, M.: Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem. Linear Algebra Appl. 431, 197\u2013210 (2009)","journal-title":"Linear Algebra Appl."},{"key":"1136_CR21","doi-asserted-by":"publisher","first-page":"638","DOI":"10.1016\/j.aml.2013.01.001","volume":"26","author":"L-L Zhang","year":"2013","unstructured":"Zhang, L.-L., Ren, Z.-R.: Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems. Appl. Math. Lett. 26, 638\u2013642 (2013)","journal-title":"Appl. Math. Lett."},{"key":"1136_CR22","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s11075-010-9416-7","volume":"57","author":"L-L 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":"1136_CR23","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 splitting iteration methods for linear complementarity problems. Numer. Algorithms. 64, 245\u2013261 (2013)","journal-title":"Numer. Algorithms."},{"key":"1136_CR24","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.cam.2013.09.079","volume":"260","author":"N Zheng","year":"2014","unstructured":"Zheng, N., Yin, J.-F.: Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an h+-matrix. J. Comput. Appl. Math. 260, 281\u2013293 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"1136_CR25","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1016\/j.aml.2013.06.015","volume":"26","author":"W Li","year":"2013","unstructured":"Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of H-matrices. Appl. Math. Lett. 26, 1159\u20131164 (2013)","journal-title":"Appl. Math. Lett."},{"key":"1136_CR26","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1016\/j.laa.2014.06.022","volume":"458","author":"W-W Xu","year":"2014","unstructured":"Xu, W.-W., Liu, H.: A modified general modulus-based matrix splitting method for linear complementarity problems with H-matrices. Linear Algebra Appl. 458, 626\u2013637 (2014)","journal-title":"Linear Algebra Appl."},{"key":"1136_CR27","doi-asserted-by":"publisher","first-page":"748","DOI":"10.1002\/nla.1985","volume":"22","author":"W-W Xu","year":"2015","unstructured":"Xu, W.-W.: Modified modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 22, 748\u2013760 (2015)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1136_CR28","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":"1136_CR29","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.laa.2015.04.029","volume":"481","author":"L-L Zhang","year":"2015","unstructured":"Zhang, L.-L., Zhang, Y.-P., Ren, Z.-R.: New convergence proofs of modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Linear Algebra Appl. 481, 83\u201393 (2015)","journal-title":"Linear Algebra Appl."},{"key":"1136_CR30","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s11075-012-9566-x","volume":"62","author":"Z-Z Bai","year":"2013","unstructured":"Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms. 62, 59\u201377 (2013)","journal-title":"Numer. Algorithms."},{"key":"1136_CR31","doi-asserted-by":"publisher","first-page":"1954","DOI":"10.1016\/j.camwa.2014.04.018","volume":"67","author":"L-T Zhang","year":"2014","unstructured":"Zhang, L.-T., Li, J.-L.: The weaker convergence of modulus-based synchronous multisplitting multi-parameters methods for linear complementarity problems. Comput. Math. Appl. 67, 1954\u20131959 (2014)","journal-title":"Comput. Math. Appl."},{"key":"1136_CR32","doi-asserted-by":"publisher","first-page":"100","DOI":"10.4208\/jcm.1403-m4195","volume":"33","author":"L-L Zhang","year":"2015","unstructured":"Zhang, L.-L.: Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems. J. Comput. Math. 33, 100\u2013112 (2015)","journal-title":"J. Comput. Math."},{"key":"1136_CR33","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.amc.2015.08.108","volume":"271","author":"Z-C Xia","year":"2015","unstructured":"Xia, Z.-C., 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."},{"key":"1136_CR34","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 iteration algorithms for a class of weakly nonlinear complementarity problems. Numer. Linear Algebra Appl. 23, 558\u2013569 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1136_CR35","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1016\/j.cam.2017.12.029","volume":"342","author":"R Li","year":"2018","unstructured":"Li, R., Yin, J.-F.: On the convergence of modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems with h+-matrices. J. Comput. Appl. Math. 342, 202\u2013209 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"1136_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.laa.2016.01.002","volume":"494","author":"S-L Xie","year":"2016","unstructured":"Xie, S.-L., Xu, X.-R., Zeng, J.-P.: Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Linear Algebra Appl. 494, 1\u201310 (2016)","journal-title":"Linear Algebra Appl."},{"key":"1136_CR37","first-page":"128","volume":"11","author":"R Li","year":"2018","unstructured":"Li, R., Wang, Y., Yin, J.-F.: On the convergence of two-step modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems with h+-matrices. Numer. Math.-Theory Me. 11, 128\u2013139 (2018)","journal-title":"Numer. Math.-Theory Me."},{"key":"1136_CR38","doi-asserted-by":"publisher","first-page":"5410","DOI":"10.1007\/s40314-018-0646-y","volume":"37","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Liu, L.: A two-step modulus-based matrix splitting iteration method for solving nonlinear complementarity problems with h+-matrices. Comput. Appl. Math. 37, 5410\u20135423 (2018)","journal-title":"Comput. Appl. Math."},{"key":"1136_CR39","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 class of restricted nonlinear complementarity problems. Numer. Algorithms. 75, 339\u2013358 (2017)","journal-title":"Numer. Algorithms."},{"key":"1136_CR40","doi-asserted-by":"publisher","first-page":"3053","DOI":"10.1007\/s40314-017-0496-z","volume":"37","author":"B-H Huang","year":"2018","unstructured":"Huang, B.-H., Ma, C.-F.: Accelerated modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Comput. Appl. Math. 37, 3053\u20133076 (2018)","journal-title":"Comput. Appl. Math."},{"key":"1136_CR41","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.amc.2019.02.015","volume":"353","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Vong, S., Liu, L.: A direct preconditioned modulus-based iteration method for sovling nonlinear complementarity problems of H-matrices. Appl. Math. Comput. 353, 396\u2013405 (2019)","journal-title":"Appl. Math. Comput."},{"key":"1136_CR42","doi-asserted-by":"publisher","first-page":"1648","DOI":"10.1080\/00207160.2018.1504928","volume":"96","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Vong, S., Liu, L.: The relaxation modulus-based matrix splitting iteration method for sovling a class of nonlinear complementarity problems. Int. J. Comput. Math. 96, 1648\u20131667 (2019)","journal-title":"Int. J. Comput. Math."},{"key":"1136_CR43","first-page":"709","volume":"12","author":"Y-J Wu","year":"2019","unstructured":"Wu, Y.-J., Yan, G.-L., Yang, A.-L.: Modulus-based synchronous multisplitting iteration methods for a restricted class of nonlinear complementarity problems. Numer. Math.-Theory Me. 12, 709\u2013726 (2019)","journal-title":"Numer. Math.-Theory Me."},{"key":"1136_CR44","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10092-018-0279-y","volume":"55","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Seakweng, V.: The modulus-based nonsmooth Newton\u2019s method for solving a class of nonlinear complementarity problems of P-matrices. Calcolo 55, 1\u201317 (2018)","journal-title":"Calcolo"},{"key":"1136_CR45","doi-asserted-by":"publisher","first-page":"480","DOI":"10.1007\/s10957-018-1361-y","volume":"180","author":"H Zheng","year":"2019","unstructured":"Zheng, H., Liu, L.: The sign-based methods for solving a class of nonlinear complementarity problems. J. Optimiz. Theory Appl. 180, 480\u2013499 (2019)","journal-title":"J. Optimiz. Theory Appl."},{"key":"1136_CR46","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/BF01581275","volume":"58","author":"L-Q Qi","year":"1993","unstructured":"Qi, L.-Q., Sun, J.: A nonsmooth version of Newton\u2019s method. Math. Program. 58, 353\u2013367 (1993)","journal-title":"Math. Program."},{"key":"1136_CR47","volume-title":"Matrix iterative analysis","author":"RS Varga","year":"1962","unstructured":"Varga, R.S.: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs (1962)"},{"key":"1136_CR48","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971262","volume-title":"Nonnegative matrices in the mathematical sciences","author":"A Berman","year":"1994","unstructured":"Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. SIAM Publisher, Philadelphia (1994)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01136-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01136-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01136-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T13:27:00Z","timestamp":1699104420000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01136-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,3]]},"references-count":48,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1136"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01136-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,3]]},"assertion":[{"value":"28 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}