{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T04:26:39Z","timestamp":1768278399825,"version":"3.49.0"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:00:00Z","timestamp":1643500800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:00:00Z","timestamp":1643500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"national natural science foundation of china","doi-asserted-by":"publisher","award":["11671220"],"award-info":[{"award-number":["11671220"]}],"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":[[2022,3]]},"DOI":"10.1007\/s10957-021-01997-7","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T20:41:05Z","timestamp":1643575265000},"page":"912-929","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Stochastic Tensor Complementarity Problem with Discrete Distribution"],"prefix":"10.1007","volume":"192","author":[{"given":"Shouqiang","family":"Du","sequence":"first","affiliation":[]},{"given":"Liyuan","family":"Cui","sequence":"additional","affiliation":[]},{"given":"Yuanyuan","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Yimin","family":"Wei","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,1,30]]},"reference":[{"key":"1997_CR1","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"1997_CR2","doi-asserted-by":"publisher","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":"1997_CR3","doi-asserted-by":"publisher","first-page":"1196","DOI":"10.1137\/S1052623497330963","volume":"10","author":"EG Birgin","year":"2000","unstructured":"Birgin, E.G., Martinez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196\u20131211 (2000)","journal-title":"SIAM J. Optim."},{"key":"1997_CR4","doi-asserted-by":"publisher","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":"1997_CR5","doi-asserted-by":"publisher","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":"1997_CR6","doi-asserted-by":"publisher","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":"1997_CR7","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/PL00011375","volume":"88","author":"B Chen","year":"2000","unstructured":"Chen, B., Chen, X., Kanzow, C.: A penalized Fischer\u2013Burmeister NCP-function. Math. Program. 88, 211\u2013216 (2000)","journal-title":"Math. Program."},{"key":"1997_CR8","doi-asserted-by":"publisher","first-page":"1022","DOI":"10.1287\/moor.1050.0160","volume":"30","author":"X Chen","year":"2005","unstructured":"Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022\u20131038 (2005)","journal-title":"Math. Oper. Res."},{"key":"1997_CR9","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s10107-007-0163-z","volume":"117","author":"X Chen","year":"2009","unstructured":"Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51\u201380 (2009)","journal-title":"Math. Program."},{"key":"1997_CR10","volume-title":"Optimization and Nonsmooth Analysis","author":"FH Clarke","year":"1983","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)"},{"key":"1997_CR11","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.laa.2018.06.028","volume":"555","author":"W Ding","year":"2018","unstructured":"Ding, W., Luo, Z., Qi, L.: P-Tensors, P$$_0$$-tensors, and their application. Linear Algebra and Its Applications 555, 336\u2013354 (2018)","journal-title":"Linear Algebra and Its Applications"},{"key":"1997_CR12","doi-asserted-by":"publisher","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":"1997_CR13","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1007\/s10957-020-01774-y","volume":"188","author":"S Du","year":"2021","unstructured":"Du, S., Ding, W., Wei, Y.: Acceptable solutions and backward errors for tensor complementarity problems. J. Optim. Theory Appl. 188, 260\u2013276 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"1997_CR14","doi-asserted-by":"publisher","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":"1997_CR15","doi-asserted-by":"publisher","first-page":"1695","DOI":"10.1007\/s11425-017-9238-6","volume":"61","author":"S Du","year":"2018","unstructured":"Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci. China Math. 61, 1695\u20131710 (2018)","journal-title":"Sci. China Math."},{"key":"1997_CR16","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1080\/02331939208843795","volume":"24","author":"A Fischer","year":"1992","unstructured":"Fischer, A.: A special Newton-type optimization method. Optimization 24, 269\u2013284 (1992)","journal-title":"Optimization"},{"key":"1997_CR17","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1007\/BF02614396","volume":"76","author":"A Fischer","year":"1997","unstructured":"Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Program. 76, 513\u2013532 (1997)","journal-title":"Math. Program."},{"key":"1997_CR18","doi-asserted-by":"publisher","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":"1997_CR19","doi-asserted-by":"publisher","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":"1997_CR20","doi-asserted-by":"publisher","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":"1997_CR21","doi-asserted-by":"publisher","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\u2013791 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1997_CR22","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1137\/S0363012994276494","volume":"35","author":"H Jiang","year":"1997","unstructured":"Jiang, H., 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."},{"key":"1997_CR23","doi-asserted-by":"publisher","unstructured":"Li, D., Chen, C., Guan, H.: A lower dimensional linear equation approach to the $${\\cal{M}}$$-tensor complementarity problem. Calcolo 58, Article 5 (2021). https:\/\/doi.org\/10.1007\/s10092-021-00397-7","DOI":"10.1007\/s10092-021-00397-7"},{"key":"1997_CR24","first-page":"87","volume":"16","author":"H Li","year":"2020","unstructured":"Li, H., Du, S., Wang, Y., Chen, M.: An inexact Levenberg\u2013Marquardt method for tensor eigenvalue complementarity problem. Pac. J. Optim. 16, 87\u201399 (2020)","journal-title":"Pac. J. Optim."},{"key":"1997_CR25","doi-asserted-by":"publisher","first-page":"1726","DOI":"10.1080\/03081087.2017.1369929","volume":"66","author":"D Liu","year":"2017","unstructured":"Liu, D., Li, W., Vong, S.W.: Tensor complementarity problem: the GUS-property and an algorithm. Linear Multilinear Algebra 66, 1726\u20131749 (2017)","journal-title":"Linear Multilinear Algebra"},{"key":"1997_CR26","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1007\/s10589-020-00222-x","volume":"77","author":"Z Ming","year":"2020","unstructured":"Ming, Z., Zhang, L., Qi, L.: Expected residual minimization method for monotone stochastic tensor complementarity problem. Comput. Optim. Appl. 77, 871\u2013896 (2020)","journal-title":"Comput. Optim. Appl."},{"key":"1997_CR27","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1007\/s10898-014-0209-8","volume":"61","author":"Q Ni","year":"2015","unstructured":"Ni, Q., Qi, L.: A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map. J. Glob. Optim. 61, 627\u2013641 (2015)","journal-title":"J. Glob. Optim."},{"key":"1997_CR28","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1287\/moor.18.1.227","volume":"18","author":"L Qi","year":"1993","unstructured":"Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227\u2013244 (1993)","journal-title":"Math. Oper. Res."},{"key":"1997_CR29","doi-asserted-by":"publisher","first-page":"1302","DOI":"10.1016\/j.jsc.2005.05.007","volume":"40","author":"L Qi","year":"2005","unstructured":"Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302\u20131324 (2005)","journal-title":"J. Symb. Comput."},{"key":"1997_CR30","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1016\/j.laa.2013.03.015","volume":"439","author":"L Qi","year":"2013","unstructured":"Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228\u2013238 (2013)","journal-title":"Linear Algebra Appl."},{"key":"1997_CR31","doi-asserted-by":"publisher","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\u2013385 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1997_CR32","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1287\/moor.22.2.301","volume":"22","author":"L Qi","year":"1997","unstructured":"Qi, L., Jiang, H.: Semismooth Karush\u2013Kuhn\u2013Tucker equations and convergence analysis of Newton and quasi-Newton methods for solving these equations. Math. Oper. Res. 22, 301\u2013325 (1997)","journal-title":"Math. Oper. Res."},{"key":"1997_CR33","doi-asserted-by":"crossref","unstructured":"Qi, L., Sun, J.: A nonsmooth version of Newton\u2019s method. Math. Program. 58, 353\u2013367 (1993)","DOI":"10.1007\/BF01581275"},{"key":"1997_CR34","volume-title":"Stochastic Programming, Handbooks in Operations Research and Management Science","year":"2003","unstructured":"Ruszczy\u0144ski, A., Shapiro, A. (eds.): Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. North-Holland, Amsterdam (2003)"},{"key":"1997_CR35","doi-asserted-by":"publisher","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":"1997_CR36","first-page":"92","volume":"03","author":"Y Song","year":"2017","unstructured":"Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 03, 92\u2013107 (2017)","journal-title":"Ann. Appl. Math."},{"key":"1997_CR37","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1007\/s10957-015-0800-2","volume":"169","author":"Y Song","year":"2016","unstructured":"Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069\u20131078 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1997_CR38","doi-asserted-by":"publisher","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":"1997_CR39","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1137\/S1052623494274970","volume":"7","author":"D Sun","year":"1997","unstructured":"Sun, D., Han, J.: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7, 463\u2013480 (1997)","journal-title":"SIAM J. Optim."},{"key":"1997_CR40","unstructured":"Tan, C., Ma, S., Dai, Y., Qian, Y.: Barzilai\u2013Borwein step size for stochastic gradient descent. In: Lee, D. D., von Luxburg, U. (eds.) NIPS\u201916: proceedings of the 30th international conference on neural information processing systems, pp. 685\u2013693 (2016)"},{"key":"1997_CR41","doi-asserted-by":"publisher","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":"1997_CR42","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1080\/10556788.2016.1180386","volume":"31","author":"Y Wang","year":"2016","unstructured":"Wang, Y., Huang, Z., Bai, X.: Exceptionally regular tensors and tensor complementarity problems. Optim. Methods Softw. 31, 815\u2013828 (2016)","journal-title":"Optim. Methods Softw."},{"key":"1997_CR43","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s10957-017-1157-5","volume":"175","author":"S Xie","year":"2017","unstructured":"Xie, S., Li, D., Xu, H.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175, 119\u2013136 (2017)","journal-title":"J. Optim. Theory Appl."},{"key":"1997_CR44","doi-asserted-by":"publisher","first-page":"2701","DOI":"10.1007\/s11590-020-01697-7","volume":"15","author":"Y Xu","year":"2021","unstructured":"Xu, Y., Huang, Z.: Bounds of the solution set of the tensor complementarity problem. Optim. Lett. 15, 2701\u20132718 (2021)","journal-title":"Optim. Lett."},{"key":"1997_CR45","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/s10957-008-9406-2","volume":"139","author":"G Zhou","year":"2008","unstructured":"Zhou, G., Caccetta, L.: Feasible semismooth Newton method for a class of stochastic linear complementarity problems. J. Optim. Theory Appl. 139, 379\u2013392 (2008)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-021-01997-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-021-01997-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-021-01997-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T08:51:46Z","timestamp":1666342306000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-021-01997-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,30]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["1997"],"URL":"https:\/\/doi.org\/10.1007\/s10957-021-01997-7","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,30]]},"assertion":[{"value":"20 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 January 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}