{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:47:02Z","timestamp":1759837622025,"version":"3.37.3"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T00:00:00Z","timestamp":1604361600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T00:00:00Z","timestamp":1604361600000},"content-version":"vor","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":"publisher","award":["11671220","11801479"],"award-info":[{"award-number":["11671220","11801479"]}],"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":["11771099"],"award-info":[{"award-number":["11771099"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2021,1]]},"DOI":"10.1007\/s10957-020-01774-y","type":"journal-article","created":{"date-parts":[[2020,11,3]],"date-time":"2020-11-03T07:02:48Z","timestamp":1604386968000},"page":"260-276","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Acceptable Solutions and Backward Errors for Tensor Complementarity Problems"],"prefix":"10.1007","volume":"188","author":[{"given":"Shouqiang","family":"Du","sequence":"first","affiliation":[]},{"given":"Weiyang","family":"Ding","sequence":"additional","affiliation":[]},{"given":"Yimin","family":"Wei","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,11,3]]},"reference":[{"key":"1774_CR1","first-page":"299","volume":"79","author":"J Pang","year":"1997","unstructured":"Pang, J.: Error bounds in mathematical programming. Math. Program. 79, 299\u2013332 (1997)","journal-title":"Math. Program."},{"key":"1774_CR2","first-page":"191","volume":"41","author":"D Klatte","year":"1995","unstructured":"Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. Z. Oper. Res. 41, 191\u2013214 (1995)","journal-title":"Z. Oper. Res."},{"key":"1774_CR3","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/BF02276920","volume":"40","author":"J Gwinner","year":"1988","unstructured":"Gwinner, J.: Acceptable solutions of linear complementarity problems. Computing 40, 361\u2013366 (1988)","journal-title":"Computing"},{"key":"1774_CR4","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1023\/A:1010982204882","volume":"101","author":"J Gwinner","year":"2001","unstructured":"Gwinner, J.: A note on backward error analysis for generalized linear complementarity problems. Ann. Oper. Res. 101, 391\u2013399 (2001)","journal-title":"Ann. Oper. Res."},{"key":"1774_CR5","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1002\/zamm.200310116","volume":"84","author":"Y Fang","year":"2004","unstructured":"Fang, Y., Huang, N.: A characterization of an acceptable solution of the extended nonlinear complementarity problem. Z. Angew. Math. Mech. 84, 564\u2013567 (2004)","journal-title":"Z. Angew. Math. Mech."},{"key":"1774_CR6","doi-asserted-by":"crossref","first-page":"933","DOI":"10.3934\/jimo.2018078","volume":"15","author":"M Zheng","year":"2019","unstructured":"Zheng, M., Zhang, Y., Huang, Z.: Global error bounds for the tensor complementarity problem with a $$P$$-tensor. J. Ind. Manag. Optim. 15, 933\u2013946 (2019)","journal-title":"J. Ind. Manag. Optim."},{"key":"1774_CR7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1080\/02331934.2017.1391254","volume":"67","author":"L Ling","year":"2018","unstructured":"Ling, L., He, H., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67, 341\u2013358 (2018)","journal-title":"Optimization"},{"key":"1774_CR8","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1007\/s10957-018-1356-8","volume":"179","author":"S Hu","year":"2018","unstructured":"Hu, S., Wang, J., Huang, Z.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 179, 983\u20131000 (2018)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR9","first-page":"3","volume":"43","author":"O Mangasarian","year":"1981","unstructured":"Mangasarian, O.: A condition number for linear inequalities and equalities. Methods Oper. Res. 43, 3\u201315 (1981)","journal-title":"Methods Oper. Res."},{"key":"1774_CR10","volume-title":"Rounding Errors in Algebraic Processes","author":"J Wilkinson","year":"1963","unstructured":"Wilkinson, J.: Rounding Errors in Algebraic Processes. NJ, Prentice-Hall, Inc, Englewood Cliffs (1963)"},{"key":"1774_CR11","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/BF01386090","volume":"6","author":"W Oettli","year":"1964","unstructured":"Oettli, W., Prager, W.: Compatibility of approximate solutions of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6, 405\u2013409 (1964)","journal-title":"Numer. Math."},{"key":"1774_CR12","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF02252346","volume":"20","author":"R Schaback","year":"1978","unstructured":"Schaback, R.: Eine rundungsgenaue formel Zur maschinellen berechnung der Prager\u2013Oettli-Schranke. Computing 20, 177\u2013182 (1978)","journal-title":"Computing"},{"key":"1774_CR13","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1007\/s11075-019-00723-9","volume":"83","author":"X Wu","year":"2020","unstructured":"Wu, X., Ke, R.: Backward errors of the linear complementarity problem. Numer. Algorithms 83, 1249\u20131257 (2020)","journal-title":"Numer. Algorithms"},{"key":"1774_CR14","volume-title":"The Linear Complementarity Problem","author":"R Cottle","year":"1992","unstructured":"Cottle, R., Pang, J., Stome, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)"},{"key":"1774_CR15","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s10957-015-0773-1","volume":"168","author":"M Che","year":"2016","unstructured":"Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475\u2013487 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR16","doi-asserted-by":"crossref","DOI":"10.1007\/978-981-15-2059-4","volume-title":"Theory and Computation of Complex Tensors and its Applications","author":"M Che","year":"2020","unstructured":"Che, M., Wei, Y.: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore (2020)"},{"key":"1774_CR17","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1007\/s10957-014-0616-5","volume":"165","author":"Y Song","year":"2015","unstructured":"Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854\u2013873 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR18","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s10589-016-9872-7","volume":"66","author":"Z Huang","year":"2017","unstructured":"Huang, Z., Qi, L.: Formulating an $$n$$-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557\u2013576 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"1774_CR19","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s10957-016-0907-0","volume":"170","author":"Y Song","year":"2016","unstructured":"Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85\u201396 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR20","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1080\/03081087.2017.1369929","volume":"66","author":"D Liu","year":"2018","unstructured":"Liu, D., Li, W., Vong, S.: Tensor complementarity problems: the GUS-property and an algorithm. Linear and Multilinear Algebra 66, 1726\u20131749 (2018)","journal-title":"Linear and Multilinear Algebra"},{"key":"1774_CR21","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1007\/s10898-018-00731-4","volume":"73","author":"S Du","year":"2019","unstructured":"Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789\u2013800 (2019)","journal-title":"J. Glob. Optim."},{"key":"1774_CR22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-019-01566-z","volume":"183","author":"Z Huang","year":"2019","unstructured":"Huang, Z., Qi, L.: Tensor complementarity problems-Part I: Basic theory. J. Optim. Theory Appl. 183, 1\u201323 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR23","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s10957-019-01568-x","volume":"183","author":"L Qi","year":"2019","unstructured":"Qi, L., Huang, Z.: Tensor complementarity problems-Part II: solution methods. J. Optim. Theory Appl. 183, 365\u2013835 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR24","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s10957-019-01573-0","volume":"183","author":"Z Huang","year":"2019","unstructured":"Huang, Z., Qi, L.: Tensor complementarity problems-Part III: applications. J. Optim. Theory Appl. 183, 771\u2013197 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR25","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s10589-019-00144-3","volume":"75","author":"S Du","year":"2020","unstructured":"Du, S., Che, M., Wei, Y.: Stochastic structured tensors to stochastic complementarity problems. Comput. Optim. Appl. 75, 649\u2013668 (2020)","journal-title":"Comput. Optim. Appl."},{"key":"1774_CR26","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1007\/s10957-019-01596-7","volume":"184","author":"V Hieu","year":"2020","unstructured":"Hieu, V., Wei, Y., Yao, J.: Notes on the optimization problems corresponding to polynomial complementarity problems. J. Optim. Theory Appl. 184, 687\u2013695 (2020)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR27","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1080\/10556788.2019.1578766","volume":"35","author":"X Wang","year":"2020","unstructured":"Wang, X., Che, M., Qi, L., Wei, Y.: Modified gradient dynamic approach to the tensor complementarity problem. Optim. Methods Softw. 35, 394\u2013415 (2020)","journal-title":"Optim. Methods Softw."},{"key":"1774_CR28","first-page":"394","volume":"35","author":"X Wang","year":"2020","unstructured":"Wang, X., Che, M., Wei, Y.: Global uniqueness and solvability of tensor complementarity problems for $$\\cal{H}_+$$-tensors. Numer. Algorithms 35, 394\u2013415 (2020)","journal-title":"Numer. Algorithms"},{"key":"1774_CR29","doi-asserted-by":"crossref","first-page":"131","DOI":"10.4208\/nmtma.OA-2018-0117","volume":"13","author":"M Che","year":"2020","unstructured":"Che, M., Qi, L., Wei, Y.: The generalized order tensor complementarity problems. Numer. Math. Theory Method Appl. 13, 131\u2013149 (2020)","journal-title":"Numer. Math. Theory Method Appl."},{"key":"1774_CR30","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s11590-018-1362-7","volume":"13","author":"M Che","year":"2019","unstructured":"Che, M., Qi, L., Wei, Y.: Stochastic $$R_0$$ tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261\u2013279 (2019)","journal-title":"Optim. Lett."},{"key":"1774_CR31","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1007\/s10957-020-01645-6","volume":"185","author":"M Zheng","year":"2020","unstructured":"Zheng, M., Huang, Z., Ma, X.: Nonemptiness and compactness of solution sets to GPCPs. J. Optim. Theory Appl. 185, 80\u201398 (2020)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR32","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0024-3795(90)90058-K","volume":"132","author":"R Mathias","year":"1990","unstructured":"Mathias, R., Pang, J.: Error bounds for the linear complementarity problem with a $$P$$-matrix. Linear Algebra Appl. 132, 123\u2013136 (1990)","journal-title":"Linear Algebra Appl."},{"key":"1774_CR33","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1287\/moor.19.4.880","volume":"19","author":"Z Luo","year":"1994","unstructured":"Luo, Z., Mangasarian, O., Ren, J., Solodov, M.: New error bounds for the linear complementarity problem. Math. Oper. Res. 19, 880\u2013892 (1994)","journal-title":"Math. Oper. Res."},{"key":"1774_CR34","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1137\/060653019","volume":"18","author":"X Chen","year":"2007","unstructured":"Chen, X., Xiang, S.: Perturbation bounds of $$P$$-matrix linear complementarity problems. SIAM J. Optim. 18, 1250\u20131265 (2007)","journal-title":"SIAM J. Optim."},{"key":"1774_CR35","first-page":"523","volume":"106","author":"X Chen","year":"2006","unstructured":"Chen, X., Xiang, S.: Computation of error bounds for $$P$$-matrix linear complementarity problems. Math. Program. 106, 523\u2013525 (2006)","journal-title":"Math. Program."},{"key":"1774_CR36","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1016\/j.laa.2010.09.049","volume":"434","author":"P Dai","year":"2011","unstructured":"Dai, P.: Error bounds for linear complementarity problems of DB-matrices. Linear Algebra Appl. 434, 830\u2013840 (2011)","journal-title":"Linear Algebra Appl."},{"key":"1774_CR37","doi-asserted-by":"crossref","unstructured":"Higham, N.: A Survey of Componentwise Perturbation Theory in Numerical Linear Algebra, In: Gautschi, W. (ed.), Mathematics of Computation 1943\u20131993: A Half Century of Computational Mathematics, volume 48 of Proceedings of Symposia in Applied Mathematics, pp. 49-77, American Mathematical Society, Providence, RI, USA (1994)","DOI":"10.1090\/psapm\/048\/1314843"},{"key":"1774_CR38","doi-asserted-by":"crossref","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":"1774_CR39","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1007\/s10915-015-0156-7","volume":"68","author":"W Ding","year":"2016","unstructured":"Ding, W., Wei, Y.: Solving multi-linear systems with $$\\cal{M}$$-tensors. J. Sci. Comput. 68, 689\u2013715 (2016)","journal-title":"J. Sci. Comput."},{"key":"1774_CR40","doi-asserted-by":"crossref","first-page":"1718","DOI":"10.1007\/s10915-018-0689-7","volume":"76","author":"H He","year":"2018","unstructured":"He, H., Ling, C., Qi, L., Zhou, G.: A globally and quadratically convergent algorithm for solving multilinear systems with $$\\cal{M}$$-tensors. J. Sci. Comput. 76, 1718\u20131741 (2018)","journal-title":"J. Sci. Comput."},{"key":"1774_CR41","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1002\/nla.2064","volume":"23","author":"W Ding","year":"2016","unstructured":"Ding, W., Hou, Z., Wei, Y.: Tensor logarithmic norm and its applicatios. Numer. Linear Algebr. 23, 989\u20131006 (2016)","journal-title":"Numer. Linear Algebr."},{"key":"1774_CR42","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s10957-016-0903-4","volume":"170","author":"X Bai","year":"2016","unstructured":"Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72\u201384 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR43","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1007\/s10957-018-1422-2","volume":"180","author":"L Han","year":"2019","unstructured":"Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949\u2013963 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1774_CR44","first-page":"308","volume":"33","author":"Y Song","year":"2017","unstructured":"Song, Y., Qi, L.: Properties of problem and some classes of structured tensors. Ann. Appl. Math. 33, 308\u2013323 (2017)","journal-title":"Ann. Appl. Math."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-020-01774-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-020-01774-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-020-01774-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,26]],"date-time":"2022-11-26T09:07:19Z","timestamp":1669453639000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-020-01774-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,3]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["1774"],"URL":"https:\/\/doi.org\/10.1007\/s10957-020-01774-y","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2020,11,3]]},"assertion":[{"value":"5 April 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}