{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T12:00:22Z","timestamp":1774353622818,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T00:00:00Z","timestamp":1540339200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11601340"],"award-info":[{"award-number":["11601340"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004733","name":"University of Macau","doi-asserted-by":"crossref","award":["MYRG2017-00098-FST"],"award-info":[{"award-number":["MYRG2017-00098-FST"]}],"id":[{"id":"10.13039\/501100004733","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100006469","name":"Macao Science and Technology Development Fund","doi-asserted-by":"crossref","award":["050\/2017\/A"],"award-info":[{"award-number":["050\/2017\/A"]}],"id":[{"id":"10.13039\/501100006469","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the Opening Project of Guangdong Provincial Engineering Technology Research Center for Data Sciences","award":["2016KF11"],"award-info":[{"award-number":["2016KF11"]}]},{"name":"Science and Technology Planning Project of Shaoguan","award":["SHAOKE [2016]44\/15"],"award-info":[{"award-number":["SHAOKE [2016]44\/15"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,10]]},"DOI":"10.1007\/s11075-018-0614-z","type":"journal-article","created":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T03:20:55Z","timestamp":1540351255000},"page":"573-592","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["A modified modulus-based matrix splitting iteration method for solving implicit complementarity problems"],"prefix":"10.1007","volume":"82","author":[{"given":"Hua","family":"Zheng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seakweng","family":"Vong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,10,24]]},"reference":[{"key":"614_CR1","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1137\/050636802","volume":"29","author":"M Alanelli","year":"2006","unstructured":"Alanelli, M., Hadjidimos, A.: A new iterative criterion for H-matrices. SIAM J. Matrix Anal. Appl. 29, 160\u2013176 (2006)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"614_CR2","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1016\/j.laa.2011.03.009","volume":"436","author":"R Bru","year":"2012","unstructured":"Bru, R., Gim\u00e9nez, I., Hadjidimos, A.: Is a \u2208 c\n                           nn a general H-matrix?. Linear Algebra Appl. 436, 364\u2013380 (2012)","journal-title":"Linear Algebra Appl."},{"key":"614_CR3","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0898-1221(96)00073-9","volume":"31","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems. Comput. Math. Appl. 31, 17\u201333 (1996)","journal-title":"Comput. Math. Appl."},{"key":"614_CR4","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/0898-1221(96)00123-X","volume":"32","author":"Z-Z Bai","year":"1996","unstructured":"Bai, Z.-Z.: New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems. Comput. Math. Appl. 32, 41\u201348 (1996)","journal-title":"Comput. Math. Appl."},{"key":"614_CR5","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/S0377-0427(98)00010-7","volume":"92","author":"Z-Z Bai","year":"1998","unstructured":"Bai, Z.-Z.: Asynchronous parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems. Appl. Math. Comput. 92, 85\u2013100 (1998)","journal-title":"Appl. Math. Comput."},{"key":"614_CR6","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/S0377-0427(98)00280-5","volume":"93","author":"Z-Z Bai","year":"1998","unstructured":"Bai, Z.-Z.: A class of asynchronous parallel nonlinear accelerated overrelaxation methods for the nonlinear complementarity problem. J. Comput. Appl. Math. 93, 35\u201344 (1998)","journal-title":"J. Comput. Appl. Math."},{"key":"614_CR7","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":"614_CR8","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":"614_CR9","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":"614_CR10","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":"614_CR11","volume-title":"Variational inequalities and quasi-variational inequalities: Applications to free boundary value problems","author":"C Baiocchi","year":"1984","unstructured":"Baiocchi, C., Capelo, A.: Variational inequalities and quasi-variational inequalities: Applications to free boundary value problems. Wiley, New York (1984)"},{"key":"614_CR12","first-page":"1189","volume":"276","author":"A Bensoussan","year":"1973","unstructured":"Bensoussan, A., Lions, J.-L.: Nouvelle formulation de probl\u00e8mes de contr\u00f4le impulsionnel et applications. C. R. Acad. Sci. Paris, Ser. A-B 276, 1189\u20131192 (1973)","journal-title":"C. R. Acad. Sci. Paris, Ser. A-B"},{"key":"614_CR13","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971262","volume-title":"Nonnegative matrix in the mathematical sciences","author":"A Berman","year":"1994","unstructured":"Berman, A., Plemmons, R. J.: Nonnegative matrix in the mathematical sciences. SIAM Publisher, Philadelphia (1994)"},{"key":"614_CR14","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1287\/moor.7.2.211","volume":"7","author":"D Chan","year":"1982","unstructured":"Chan, D., Pang, J.-S.: The generalized quasi-variational problem. Math. Oper. Res. 7, 211\u2013222 (1982)","journal-title":"Math. Oper. Res."},{"key":"614_CR15","volume-title":"Variational inequalities and complementarity problems: Theory and applications","author":"RW Cottle","year":"1980","unstructured":"Cottle, R.W., Giannessi, F., Lions, J.-L.: Variational inequalities and complementarity problems: Theory and applications. Wiley, Chichester (1980)"},{"key":"614_CR16","volume-title":"The Linear Complementarity Problem","author":"RW Cottle","year":"1992","unstructured":"Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic, San Diego (1992)"},{"key":"614_CR17","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 16, 129\u2013143 (2009)","journal-title":"Numer. Linear Algebra"},{"key":"614_CR18","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/0024-3795(89)90074-8","volume":"119","author":"A Frommer","year":"1989","unstructured":"Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141\u2013152 (1989)","journal-title":"Linear Algebra Appl."},{"key":"614_CR19","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":"614_CR20","doi-asserted-by":"publisher","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Steifel, E.: Methods of conjugate gradient for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409\u2013436 (1952)","journal-title":"J. Res. Natl. Bur. Stand."},{"key":"614_CR21","doi-asserted-by":"publisher","unstructured":"Hong, J.-T., Li, C.-L.: Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems. Numer. Linear Algebra, \n                    https:\/\/doi.org\/10.1002\/nla.2044\n                    \n                   (2016)","DOI":"10.1002\/nla.2044"},{"key":"614_CR22","first-page":"272","volume":"4","author":"J-G Hu","year":"1982","unstructured":"Hu, J.-G.: Estimates of \n                    \n                      \n                    \n                    \n                      \n                        |\n                        |\n                        \n                          \n                            b\n                          \n                          \n                            \u2212\n                            1\n                          \n                        \n                        c\n                        |\n                        \n                          \n                            |\n                          \n                          \n                            \u221e\n                          \n                        \n                      \n                    \n                    $||{b^{-1}c}||_{\\infty }$\n                   and their applications. Math. Numer. Sin. 4, 272\u2013282 (1982)","journal-title":"Math. Numer. Sin."},{"key":"614_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0084653","volume-title":"Complementarity problems, lecture notes in mathematics","author":"G Isac","year":"1992","unstructured":"Isac, G.: Complementarity problems, lecture notes in mathematics. Springer, New York (1992)"},{"key":"614_CR24","volume-title":"Leray-schauder type alternatives, complementarity problems and variational inequalities","author":"G Isac","year":"2006","unstructured":"Isac, G.: Leray-schauder type alternatives, complementarity problems and variational inequalities. Springer, Boston (2006)"},{"key":"614_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":"614_CR26","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":"614_CR27","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10092-015-0143-2","volume":"53","author":"S-M Liu","year":"2016","unstructured":"Liu, S.-M., Zheng, H., Li, W.: A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. Calcolo 53, 189\u2013199 (2016)","journal-title":"Calcolo"},{"key":"614_CR28","volume-title":"Implicit variational problems and quasi-variational inequalities, lecture notes in mathematics","author":"U Mosco","year":"1976","unstructured":"Mosco, U.: Implicit variational problems and quasi-variational inequalities, lecture notes in mathematics. Springer, New York (1976)"},{"key":"614_CR29","volume-title":"On some nonlinear quasi variational inequalities and implicit complementarity problems in stochastic control theory in variational inequalities and complementarity problems","author":"U Mosco","year":"1980","unstructured":"Mosco, U.: On some nonlinear quasi variational inequalities and implicit complementarity problems in stochastic control theory in variational inequalities and complementarity problems. Wiley, New York (1980)"},{"key":"614_CR30","volume-title":"Linear complementarity, linear and nonlinear programming","author":"KG Murty","year":"1988","unstructured":"Murty, K.G.: Linear complementarity, linear and nonlinear programming. Heldermann Verlag, Berlin (1988)"},{"key":"614_CR31","first-page":"99","volume":"24","author":"M Noor","year":"1982","unstructured":"Noor, M.: Mildly nonlinear variational inequalities. Mathematica 24, 99\u2013110 (1982)","journal-title":"Mathematica"},{"issue":"3","key":"614_CR32","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/0893-9659(89)90063-3","volume":"2","author":"M Noor","year":"1989","unstructured":"Noor, M.: Nonlinear quasi complementarity problems. Appl. Math. Lett. 2(3), 251\u2013254 (1989)","journal-title":"Appl. Math. Lett."},{"key":"614_CR33","first-page":"249","volume":"27","author":"M Noor","year":"1985","unstructured":"Noor, M.: Linear quasi-complementarity problem. Utilitas Mathematica 27, 249\u2013260 (1985)","journal-title":"Utilitas Mathematica"},{"key":"614_CR34","first-page":"344","volume":"130","author":"M Noor","year":"1988","unstructured":"Noor, M.: The quasi-complementarity problem. J. Optimiz. Theory Appl. 130, 344\u2013353 (1988)","journal-title":"J. Optimiz. Theory Appl."},{"key":"614_CR35","first-page":"437","volume":"133","author":"M Noor","year":"1988","unstructured":"Noor, M.: Fixed point approach for complementarity problems. J. Optimiz. Theory Appl. 133, 437\u2013448 (1988)","journal-title":"J. Optimiz. Theory Appl."},{"key":"614_CR36","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF00934765","volume":"37","author":"J-S Pang","year":"1982","unstructured":"Pang, J.-S.: On the convergence of a basic iterative method for the implicit complementarity problems. J. Optimiz. Theory Appl. 37, 149\u2013162 (1982)","journal-title":"J. Optimiz. Theory Appl."},{"key":"614_CR37","volume-title":"The implicit complemenarity problem","author":"J-S Pang","year":"1992","unstructured":"Pang, J.-S.: The implicit complemenarity problem. Academic Press, San Diego (1992)"},{"key":"614_CR38","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856\u2013869 (1986)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"614_CR39","volume-title":"Piecewise-linear modelling and analysis","author":"WMG Bokhoven van","year":"1981","unstructured":"van Bokhoven, W.M.G.: Piecewise-linear modelling and analysis. Proefschrift, Eindhoven (1981)"},{"key":"614_CR40","first-page":"11","volume":"31","author":"Q Yuan","year":"2009","unstructured":"Yuan, Q, Yin, H.-Y.: Stationary point of the minimization reformulations of implicit complementarity problems. Numer. Math. A J. Chinese Universities 31, 11\u201318 (2009)","journal-title":"Numer. Math. A J. Chinese Universities"},{"key":"614_CR41","first-page":"551","volume":"23","author":"W-P Zhan","year":"2000","unstructured":"Zhan, W.-P., Zhou, S.-Z., Zeng, J.-P.: Iterative methods for a kind of quasi-complementarity problems. Acta. Math. Appl. Sin. 23, 551\u2013556 (2000)","journal-title":"Acta. Math. Appl. Sin."},{"key":"614_CR42","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 for linear complementarity problems. Numer. Algorithms 57, 83\u201399 (2011)","journal-title":"Numer. Algorithms"},{"key":"614_CR43","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/s10957-013-0362-0","volume":"160","author":"L-L Zhang","year":"2014","unstructured":"Zhang, L.-L.: Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems. J. Optimiz. Theory Appl. 160, 189\u2013203 (2014)","journal-title":"J. Optimiz. Theory Appl."},{"key":"614_CR44","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":"614_CR45","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":"614_CR46","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":"614_CR47","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. Algorithms 74, 137\u2013152 (2017)","journal-title":"Numer. Algorithms"},{"key":"614_CR48","doi-asserted-by":"publisher","unstructured":"Zheng, H., Vong, S.: Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems. Linear Multilinear Algebra \n                    https:\/\/doi.org\/10.1080\/03081087.2018.1470602\n                    \n                   (2018)","DOI":"10.1080\/03081087.2018.1470602"},{"key":"614_CR49","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\u2013262 (2013)","journal-title":"Numer. Algorithms"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0614-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-018-0614-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0614-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,23]],"date-time":"2019-10-23T20:13:19Z","timestamp":1571861599000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-018-0614-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,24]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,10]]}},"alternative-id":["614"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0614-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,24]]},"assertion":[{"value":"29 November 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}