{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T12:14:44Z","timestamp":1768738484471,"version":"3.49.0"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"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 Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001211"],"award-info":[{"award-number":["12001211"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071159"],"award-info":[{"award-number":["12071159"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1811464"],"award-info":[{"award-number":["U1811464"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071159"],"award-info":[{"award-number":["12071159"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171168"],"award-info":[{"award-number":["12171168"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Fujian Province, China","award":["2022J01194"],"award-info":[{"award-number":["2022J01194"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2023,9]]},"DOI":"10.1007\/s10589-023-00482-3","type":"journal-article","created":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T18:02:36Z","timestamp":1682359356000},"page":"345-381","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A smoothing Newton method based on the modulus equation for a class of weakly nonlinear complementarity problems"],"prefix":"10.1007","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5011-0801","authenticated-orcid":false,"given":"Baohua","family":"Huang","sequence":"first","affiliation":[]},{"given":"Wen","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"482_CR1","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1137\/S0036144595285963","volume":"39","author":"M Ferri","year":"1997","unstructured":"Ferri, M., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669\u2013713 (1997)","journal-title":"SIAM Rev."},{"key":"482_CR2","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1287\/mnsc.11.7.681","volume":"11","author":"CE Lemke","year":"1965","unstructured":"Lemke, C.E.: Bimatrix equilibrium points and methematical programming. Manage. Sci. 11, 681\u2013689 (1965)","journal-title":"Manage. Sci."},{"key":"482_CR3","volume-title":"Piecewise-Linear Modelling and Analysis","author":"W Van Bokhoven","year":"1981","unstructured":"Van Bokhoven, W.: Piecewise-Linear Modelling and Analysis. Proefschrift, Eindhoven (1981)"},{"key":"482_CR4","volume-title":"Linear Complementarity, Linear and Nonlinear Programming","author":"KG Murty","year":"1988","unstructured":"Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)"},{"key":"482_CR5","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1002\/nla.609","volume":"16","author":"JL 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":"482_CR6","doi-asserted-by":"crossref","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":"482_CR7","doi-asserted-by":"crossref","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":"482_CR8","doi-asserted-by":"crossref","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 problem. Numer. Algor. 64, 245\u2013262 (2013)","journal-title":"Numer. Algor."},{"key":"482_CR9","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.cam.2017.12.028","volume":"336","author":"A Hadjidimos","year":"2018","unstructured":"Hadjidimos, A., Zhang, L.L.: Comparison of three classes of algorithms for the solution of the linear complementarity problem with an $$H_+$$-matrix. J. Comput. Appl. Math. 336, 175\u2013191 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"482_CR10","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0168-9274(02)00233-7","volume":"45","author":"MD Koulisianis","year":"2003","unstructured":"Koulisianis, M.D., Papatheodorou, T.S.: Improving projected successive overrelaxation method for linear complementarity problems. Appl. Numer. Math. 45, 29\u201344 (2003)","journal-title":"Appl. Numer. Math."},{"key":"482_CR11","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.amc.2006.09.067","volume":"188","author":"Y Li","year":"2007","unstructured":"Li, Y., Dai, P.: Generalized AOR methods for linear complementarity problem. Appl. Math. Comput. 188, 7\u201318 (2007)","journal-title":"Appl. Math. Comput."},{"key":"482_CR12","first-page":"9","volume":"29","author":"L Liao","year":"1999","unstructured":"Liao, L., Qi, H.: A neural network for the linear complementarity problem. Math. Comput. Model. 29, 9\u201318 (1999)","journal-title":"Math. Comput. Model."},{"key":"482_CR13","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1137\/S1052623498335080","volume":"11","author":"X Chen","year":"2000","unstructured":"Chen, X., Ye, Y.: On smoothing methods for the $$P_0$$ matrix linear complementarity problem. SIAM J. Optim. 11, 341\u2013363 (2000)","journal-title":"SIAM J. Optim."},{"key":"482_CR14","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF01582255","volume":"48","author":"PT Harker","year":"1990","unstructured":"Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161\u2013220 (1990)","journal-title":"Math. Program."},{"key":"482_CR15","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-Verlag, New York (2003)"},{"key":"482_CR16","first-page":"247","volume":"23","author":"HD Qi","year":"2001","unstructured":"Qi, H.D., Zhang, Y.Z.: A smoothing Newton method for complementarity problems. Math. Numer. Sin. 23, 247\u2013264 (2001). (in Chinese)","journal-title":"Math. Numer. Sin."},{"key":"482_CR17","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.amc.2015.08.108","volume":"271","author":"ZC 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":"482_CR18","doi-asserted-by":"crossref","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, 558\u2013569 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"key":"482_CR19","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.apnum.2016.05.004","volume":"108","author":"CF Ma","year":"2016","unstructured":"Ma, C.F., Huang, N.: Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems. Appl. Numer. Math. 108, 116\u2013124 (2016)","journal-title":"Appl. Numer. Math."},{"key":"482_CR20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.laa.2016.01.002","volume":"494","author":"SL Xie","year":"2016","unstructured":"Xie, S.L., Xu, H.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":"482_CR21","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s42967-020-00074-6","volume":"3","author":"G Wang","year":"2021","unstructured":"Wang, G., Tan, F.: Modulus-based multisplitting iteration method for a class of weakly nonlinear complementarity problem. Com. Appl. Math. Comput. 3, 419\u2013427 (2021)","journal-title":"Com. Appl. Math. Comput."},{"key":"482_CR22","doi-asserted-by":"crossref","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":"482_CR23","doi-asserted-by":"crossref","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 solving a class of nonlinear complementarity problems. Int. J. Comput. Math. 96, 1648\u20131667 (2019)","journal-title":"Int. J. Comput. Math."},{"key":"482_CR24","doi-asserted-by":"crossref","first-page":"3053","DOI":"10.1007\/s40314-017-0496-z","volume":"37","author":"BH 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":"482_CR25","doi-asserted-by":"crossref","first-page":"6795","DOI":"10.1007\/s40314-018-0687-2","volume":"37","author":"YF Ke","year":"2018","unstructured":"Ke, Y.F., Ma, C.F., Zhang, H.: The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems. Comput. Appl. Math. 37, 6795\u20136820 (2018)","journal-title":"Comput. Appl. Math."},{"key":"482_CR26","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 solving nonlinear complementarity problems of $$H$$-matrices. Appl. Math. Comput. 353, 396\u2013405 (2019)","journal-title":"Appl. Math. Comput."},{"key":"482_CR27","doi-asserted-by":"crossref","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, 1481\u20131490 (2017)","journal-title":"Calcolo"},{"key":"482_CR28","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1287\/moor.17.3.691","volume":"17","author":"S Robinson","year":"1992","unstructured":"Robinson, S.: Normal maps induced by linear transformation. Math. Oper. Res. 17, 681\u2013714 (1992)","journal-title":"Math. Oper. Res."},{"key":"482_CR29","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1137\/S1052623494279110","volume":"7","author":"F Facchinei","year":"1997","unstructured":"Facchinei, F., Soares, J.: A new metrit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225\u2013247 (1997)","journal-title":"SIAM J. Optim."},{"key":"482_CR30","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10092-018-0279-y","volume":"55","author":"H Zheng","year":"2018","unstructured":"Zheng, H., Vong, S.: The modulus-based nonsmooth Newton\u2019s method for solving a class of nonlinear complementarity problems of $$P$$-matrices. Calcolo 55, 37 (2018)","journal-title":"Calcolo"},{"key":"482_CR31","doi-asserted-by":"crossref","DOI":"10.1016\/j.amc.2020.125175","volume":"387","author":"X Zhang","year":"2020","unstructured":"Zhang, X., Peng, Z.: A modulus-based nonmonotone line search method for nonlinear complementarity problems. Appl. Math. Comput. 387, 125175 (2020)","journal-title":"Appl. Math. Comput."},{"key":"482_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s101079900127","volume":"87","author":"L Qi","year":"2000","unstructured":"Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1\u201335 (2000)","journal-title":"Math. Program."},{"key":"482_CR33","doi-asserted-by":"crossref","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":"482_CR34","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971309","volume-title":"Optimization and Nonsmooth Analysis","author":"FH Clarke","year":"1990","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)"},{"key":"482_CR35","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/BF01581275","volume":"58","author":"L Qi","year":"1993","unstructured":"Qi, L., Sun, J.: A nonsmooth version of Newton\u2019s method. Math. Program. 58, 353\u2013367 (1993)","journal-title":"Math. Program."},{"key":"482_CR36","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1023\/A:1022969507994","volume":"25","author":"L Qi","year":"2003","unstructured":"Qi, L., Yin, H.X.: A strongly semismooth integral function and its application. Comput. Optim. Appl. 25, 223\u2013246 (2003)","journal-title":"Comput. Optim. Appl."},{"key":"482_CR37","doi-asserted-by":"crossref","first-page":"3739","DOI":"10.1016\/j.na.2010.01.012","volume":"72","author":"JS Chen","year":"2010","unstructured":"Chen, J.S., Pan, S.H., Lin, T.C.: A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs. Nonlinear Anal. 72, 3739\u20133758 (2010)","journal-title":"Nonlinear Anal."},{"key":"482_CR38","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10589-009-9315-9","volume":"50","author":"JS Chen","year":"2011","unstructured":"Chen, J.S., Huang, Z.H., She, C.Y.: A new class of penalized NCP-functions and its properties. Comput. Optim. Appl. 50, 49\u201373 (2011)","journal-title":"Comput. Optim. Appl."},{"key":"482_CR39","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.cam.2008.10.056","volume":"230","author":"SL Hu","year":"2009","unstructured":"Hu, S.L., Huang, Z.H., Chen, J.S.: Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems. J. Comput. Appl. Math. 230, 69\u201382 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"482_CR40","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.apnum.2017.04.012","volume":"120","author":"XH Miao","year":"2017","unstructured":"Miao, X.H., Yang, J.T., Saheya, B., Chen, J.S.: A smoothing Newton method for absolute value equation associated with second-order cone. Appl. Numer. Math. 120, 82\u201396 (2017)","journal-title":"Appl. Numer. Math."},{"key":"482_CR41","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/0614081","volume":"14","author":"B Chen","year":"1993","unstructured":"Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14, 1168\u20131190 (1993)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"482_CR42","doi-asserted-by":"crossref","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":"482_CR43","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF02591991","volume":"36","author":"OL Mangasarian","year":"1986","unstructured":"Mangasarian, O.L., Shiau, T.H.: Error bounds for monotone linear complementarity problems. Math. Programm. 36, 81\u201389 (1986)","journal-title":"Math. Programm."},{"key":"482_CR44","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10107-007-0163-z","volume":"117","author":"XJ Chen","year":"2009","unstructured":"Chen, X.J., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program Ser. B 117, 51\u201380 (2009)","journal-title":"Math. Program Ser. B"},{"key":"482_CR45","first-page":"49","volume":"1","author":"OL Mangasarian","year":"1994","unstructured":"Mangasarian, O.L., Ren, J.: New Error Bounds for the Nonlinear Complementarity Problems. Comm. Appl. Nonlinear Anal. 1, 49\u201356 (1994)","journal-title":"Comm. Appl. Nonlinear Anal."},{"key":"482_CR46","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1093\/imanum\/16.1.31","volume":"16","author":"KH Hoffmann","year":"1996","unstructured":"Hoffmann, K.H., Zou, J.: Parallel solution of variational inequality problems with nonlinear source terms. IMA J. Numer. Anal. 16, 31\u201345 (1996)","journal-title":"IMA J. Numer. Anal."},{"key":"482_CR47","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1137\/S0363012994276494","volume":"35","author":"HY Jiang","year":"1997","unstructured":"Jiang, H.Y., Qi, L.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control. Optim. 35, 178\u2013193 (1997)","journal-title":"SIAM J. Control. Optim."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00482-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-023-00482-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00482-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,11]],"date-time":"2023-12-11T10:26:26Z","timestamp":1702290386000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-023-00482-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,24]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["482"],"URL":"https:\/\/doi.org\/10.1007\/s10589-023-00482-3","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,24]]},"assertion":[{"value":"7 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2023","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 authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}