{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T13:55:42Z","timestamp":1780408542343,"version":"3.54.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"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":["12201275"],"award-info":[{"award-number":["12201275"]}],"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":["12131004"],"award-info":[{"award-number":["12131004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2021J01661"],"award-info":[{"award-number":["2021J01661"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ministry of Science and Technology of China","award":["2021YFA1003600"],"award-info":[{"award-number":["2021YFA1003600"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s10957-024-02529-9","type":"journal-article","created":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T07:02:54Z","timestamp":1728370974000},"page":"714-744","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Scalable Relaxation Two-Sweep Modulus-Based Matrix Splitting Methods for Vertical LCP"],"prefix":"10.1007","volume":"203","author":[{"given":"Dongmei","family":"Yu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huiling","family":"Wei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cairong","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Deren","family":"Han","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,8]]},"reference":[{"key":"2529_CR1","volume-title":"Iterative Solution Methods","author":"O Axelsson","year":"1996","unstructured":"Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1996)"},{"key":"2529_CR2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1080\/00207169708804569","volume":"63","author":"ZZ 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":"2529_CR3","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":"2529_CR4","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, Philadelphia (1994)"},{"key":"2529_CR5","unstructured":"Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)"},{"key":"2529_CR6","first-page":"542","volume":"348","author":"PF Dai","year":"2019","unstructured":"Dai, P.F., Li, J.C., Bai, J.C., Qi, J.M.: A preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problem. Appl. Math. Comput. 348, 542\u2013551 (2019)","journal-title":"Appl. Math. Comput."},{"key":"2529_CR7","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/BF01585560","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":"2529_CR8","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 $$P_0$$-matrices. Optim. Methods Softw. 10, 747\u2013761 (1999)","journal-title":"Optim. Methods Softw."},{"key":"2529_CR9","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":"2529_CR10","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":"2529_CR11","doi-asserted-by":"publisher","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":"2529_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108344","volume":"134","author":"JW He","year":"2022","unstructured":"He, J.W., Vong, S.: 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":"2529_CR13","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1002\/nla.2044","volume":"23","author":"JT Hong","year":"2016","unstructured":"Hong, J.T., Li, C.L.: Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems. Numer. Linear Algebra Appl. 23, 629\u2013641 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2529_CR14","first-page":"272","volume":"3","author":"JG Hu","year":"1982","unstructured":"Hu, J.G.: Estimates of $$\\Vert B^{-1}C\\Vert _\\infty $$ and their applications. Math. Numer. Sin. 3, 272\u2013282 (1982)","journal-title":"Math. Numer. Sin."},{"key":"2529_CR15","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, 558\u2013569 (2016)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2529_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, 994\u20131006 (1996)","journal-title":"SIAM J. Optim."},{"key":"2529_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, 181\u2013190 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2529_CR18","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF02592211","volume":"74","author":"SR Mohan","year":"1996","unstructured":"Mohan, S.R., Neogy, S.K., Sridhar, R.: The generalized linear complementarity problem revisited. Math. Program. 74, 197\u2013218 (1996)","journal-title":"Math. Program."},{"key":"2529_CR19","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":"2529_CR20","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":"2529_CR21","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. Algorithms 83, 201\u2013219 (2020)","journal-title":"Numer. Algorithms"},{"key":"2529_CR22","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":"2529_CR23","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1016\/0024-3795(87)90125-X","volume":"88\u201389","author":"M Neumann","year":"1987","unstructured":"Neumann, M., Plemmons, R.J.: Convergence of parallel multisplitting iterative methods for M-matrices. Linear Algebra Appl. 88\u201389, 559\u2013573 (1987)","journal-title":"Linear Algebra Appl."},{"key":"2529_CR24","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1137\/0704017","volume":"4","author":"JM Ortega","year":"1967","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Monotone iterations for nonlinear equations with applications to Gauss-Seidel methods. SIAM J. Numer. Anal. 4, 171\u2013196 (1967)","journal-title":"SIAM J. Numer. Anal."},{"key":"2529_CR25","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s101070050104","volume":"86","author":"JM Peng","year":"1999","unstructured":"Peng, J.M., Lin, Z.H.: A non-interior continuation method for generalized linear complementarity problems. Math. Program. 86, 533\u2013563 (1999)","journal-title":"Math. Program."},{"key":"2529_CR26","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.1016\/j.camwa.2006.02.006","volume":"51","author":"SQ Shen","year":"2006","unstructured":"Shen, S.Q., Huang, T.Z.: Convergence and comparison theorems for double splittings of matrices. Comput. Math. Appl. 51, 1751\u20131760 (2006)","journal-title":"Comput. Math. Appl."},{"key":"2529_CR27","doi-asserted-by":"publisher","first-page":"1882","DOI":"10.3390\/sym14091882","volume":"14","author":"YL Song","year":"2022","unstructured":"Song, Y.L., Zheng, H., Lu, X.P., Vong, S.: A two-step iteration method for vertical linear complementarity problems. Symmetry 14, 1882 (2022)","journal-title":"Symmetry"},{"key":"2529_CR28","doi-asserted-by":"publisher","first-page":"474","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, 474\u2013485 (1989)","journal-title":"J. Math. Anal. Appl."},{"key":"2529_CR29","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1016\/0024-3795(93)00184-2","volume":"223\u2013224","author":"R Sznajder","year":"1995","unstructured":"Sznajder, R., Gowda, M.S.: Generalizations of $$P_0$$-and $$P$$-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223\u2013224, 695\u2013715 (1995)","journal-title":"Linear Algebra Appl."},{"key":"2529_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115430","volume":"437","author":"D Wang","year":"2024","unstructured":"Wang, D., Li, J.C.: Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problem. J. Comput. Appl. Math. 437, 115430 (2024)","journal-title":"J. Comput. Appl. Math."},{"key":"2529_CR31","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1016\/j.cam.2008.05.022","volume":"226","author":"L Wang","year":"2009","unstructured":"Wang, L., Song, Y.Z.: Preconditioned AOR iterative methods for $$M$$-matrices. J. Comput. Appl. Math. 226, 114\u2013124 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"2529_CR32","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/j.cam.2016.02.011","volume":"302","author":"SL Wu","year":"2016","unstructured":"Wu, S.L., Li, C.X.: Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems. J. Comput. Appl. Math. 302, 327\u2013339 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"2529_CR33","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.1007\/s11590-021-01781-6","volume":"16","author":"SL Wu","year":"2022","unstructured":"Wu, S.L., Li, C.X.: A class of new modulus-based matrix splitting methods for linear complementarity problem. Optim. Lett. 16, 1427\u20131443 (2022)","journal-title":"Optim. Lett."},{"key":"2529_CR34","first-page":"34","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":"2529_CR35","doi-asserted-by":"publisher","first-page":"1377","DOI":"10.1007\/s11075-019-00660-7","volume":"82","author":"C Yang","year":"2019","unstructured":"Yang, C., Wang, A.: Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems. Numer. Algorithms 82, 1377\u20131394 (2019)","journal-title":"Numer. Algorithms"},{"key":"2529_CR36","doi-asserted-by":"publisher","first-page":"13368","DOI":"10.3934\/math.2023677","volume":"8","author":"DM Yu","year":"2023","unstructured":"Yu, D.M., Zhang, Y.M., Chen, C.R., Han, D.R.: A new relaxed acceleration two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems. AIMS Math. 8, 13368\u201313389 (2023)","journal-title":"AIMS Math."},{"key":"2529_CR37","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1007\/BF02437876","volume":"24","author":"LP Zhang","year":"2003","unstructured":"Zhang, L.P., Gao, Z.Y.: Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems. Appl. Math. Mech. 24, 738\u2013746 (2003)","journal-title":"Appl. Math. Mech."},{"key":"2529_CR38","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. Algorithms 82, 573\u2013592 (2019)","journal-title":"Numer. Algorithms"},{"key":"2529_CR39","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s11075-016-0142-7","volume":"74","author":"H Zheng","year":"2016","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 (2016)","journal-title":"Numer. Algorithms"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02529-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02529-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02529-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T18:06:29Z","timestamp":1730484389000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02529-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["2529"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02529-9","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10]]},"assertion":[{"value":"4 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}