{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T10:55:23Z","timestamp":1778324123764,"version":"3.51.4"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2023,9]]},"DOI":"10.1007\/s11590-023-01977-y","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T20:06:46Z","timestamp":1676664406000},"page":"1697-1709","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Existence of the least element solution of the vertical block Z-tensor complementarity problem"],"prefix":"10.1007","volume":"17","author":[{"given":"Ruoke","family":"Meng","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2269-961X","authenticated-orcid":false,"given":"Zheng-Hai","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Yong","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,17]]},"reference":[{"issue":"1","key":"1977_CR1","doi-asserted-by":"publisher","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. Theor. Meth. Appl. 13(1), 131\u2013149 (2020)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"1977_CR2","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 Press, Boston (1992)"},{"key":"1977_CR3","unstructured":"Han, J., Xiu, N., Qi, H.D.: Nonlinear complementarity theory and algorithms. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)"},{"issue":"1","key":"1977_CR4","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/S0021-9800(70)80010-2","volume":"8","author":"RW Cottle","year":"1970","unstructured":"Cottle, R.W., Dantzig, G.B.: A generalization of the linear complementarity problem. J. Comb. Theory 8(1), 79\u201390 (1970)","journal-title":"J. Comb. Theory"},{"issue":"2","key":"1977_CR5","doi-asserted-by":"publisher","first-page":"1421","DOI":"10.1016\/j.jmaa.2006.07.036","volume":"329","author":"AA Ebiefung","year":"2007","unstructured":"Ebiefung, A.A.: Existence theory and $$Q$$-matrix characterization for the generalized linear complementarity problem: Revisited. J. Math. Anal. Appl. 329(2), 1421\u20131429 (2007)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1977_CR6","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF02061065","volume":"44","author":"AA Ebiefung","year":"1993","unstructured":"Ebiefung, A.A., Kostreva, M.M.: The generalized Leontief input-output model and its application to the choice of new technology. Ann. Oper. Res. 44(2), 161\u2013172 (1993)","journal-title":"Ann. Oper. Res."},{"issue":"2","key":"1977_CR7","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1023\/A:1008224418676","volume":"11","author":"AA Ebiefung","year":"1997","unstructured":"Ebiefung, A.A., Kostreva, M.M.: The generalized linear complementarity problem: Least element theory and $$Z$$-matrices. J. Global Optim. 11(2), 151\u2013161 (1997)","journal-title":"J. Global Optim."},{"issue":"3","key":"1977_CR8","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1137\/S0895479892237859","volume":"15","author":"MS Gowda","year":"1994","unstructured":"Gowda, M.S., Sznajder, R.: The generalized order linear complementarity problem. SIAM J. Matrix Anal. Appl. 15(3), 779\u2013795 (1994)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"1977_CR9","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(4), 994\u20131006 (1996)","journal-title":"SIAM J. Optim."},{"key":"1977_CR10","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1007\/s101070050108","volume":"86","author":"SR Mohan","year":"1999","unstructured":"Mohan, S.R., Neogy, S.K., Parthasarathy, T., Sinha, S.: Vertical linear complementarity and discounted zero-sum stochastic games with ARAT structure. Math. Program. 86, 637\u2013648 (1999)","journal-title":"Math. Program."},{"issue":"1","key":"1977_CR11","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1137\/S0895479897329837","volume":"21","author":"HD Qi","year":"1999","unstructured":"Qi, H.D., Liao, L.Z.: A smoothing Newton method for extended vertical linear complementarity problems. SIAM J. Matrix Anal. Appl. 21(1), 45\u201366 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1977_CR12","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/s101070050104","volume":"86","author":"JM Peng","year":"1999","unstructured":"Peng, J.M., Lin, L.: A non-interior continuation method for generalized linear complementarity problems. Math. Program. 86, 533\u2013563 (1999)","journal-title":"Math. Program."},{"key":"1977_CR13","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."},{"issue":"3","key":"1977_CR14","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1007\/s10589-016-9872-7","volume":"66","author":"ZH Huang","year":"2017","unstructured":"Huang, Z.H., Qi, L.: Formulating an $$n$$-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66(3), 557\u2013576 (2017)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"1977_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10957-019-01566-z","volume":"183","author":"ZH Huang","year":"2019","unstructured":"Huang, Z.H., Qi, L.: Tensor complementarity problems - part I: basic theory. J. Optim. Theory Appl. 183(1), 1\u201323 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1977_CR16","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.H.: Tensor complementarity problems - part II: solution methods. J. Optim. Theory Appl. 183(2), 365\u2013385 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1977_CR17","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1007\/s10957-019-01573-0","volume":"183","author":"ZH Huang","year":"2019","unstructured":"Huang, Z.H., Qi, L.: Tensor complementarity problems - part III: applications. J. Optim. Theory Appl. 183(3), 771\u2013791 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"1977_CR18","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s11590-016-1013-9","volume":"11","author":"Z Luo","year":"2017","unstructured":"Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to $$Z$$-tensor complementarity problems. Optim. Lett. 11, 471\u2013482 (2017)","journal-title":"Optim. Lett."},{"key":"1977_CR19","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s10957-017-1157-5","volume":"175","author":"SL Xie","year":"2017","unstructured":"Xie, S.L., Li, D.H., Xu, H.R.: 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":"1977_CR20","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1007\/s11590-018-1268-4","volume":"13","author":"HR Xu","year":"2019","unstructured":"Xu, H.R., Li, D.H., Xie, S.L.: An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor. Optim. Lett. 13, 685\u2013694 (2019)","journal-title":"Optim. Lett."},{"key":"1977_CR21","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1016\/j.laa.2019.09.025","volume":"584","author":"SL Xie","year":"2020","unstructured":"Xie, S.L., Xu, H.R.: A two-level additive Schwarz method for a kind of tensor complementarity problem. Linear Algebra Appl. 584, 394\u2013408 (2020)","journal-title":"Linear Algebra Appl."},{"key":"1977_CR22","volume-title":"Finding the least element of a nonnegative solution set of a class of polynomial inequalities","author":"ZH Huang","year":"2022","unstructured":"Huang, Z.H., Li, Y.F., Miao, X.H.: Finding the least element of a nonnegative solution set of a class of polynomial inequalities. Accepted for publication in SIAM J. Matrix Anal, Appl (2022)"},{"key":"1977_CR23","doi-asserted-by":"publisher","first-page":"796","DOI":"10.1137\/S1052623400366802","volume":"11","author":"JB Lasserre","year":"2001","unstructured":"Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796\u2013817 (2001)","journal-title":"SIAM J. Optim."},{"issue":"4\u20135","key":"1977_CR24","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1080\/10556780802699201","volume":"24","author":"D Henrion","year":"2009","unstructured":"Henrion, D., Lasserre, J.-B., L\u00f6fberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Method. Softw. 24(4\u20135), 761\u2013779 (2009)","journal-title":"Optim. Method. Softw."},{"key":"1977_CR25","doi-asserted-by":"crossref","unstructured":"Strum, J.F.: Using sedumi $$1.02$$, a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11-12, 625-653 (1999)","DOI":"10.1080\/10556789908805766"},{"key":"1977_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-022-01263-8","author":"ZH Huang","year":"2022","unstructured":"Huang, Z.H., Li, Y.F., Wang, Y.: A fixed point iterative method for tensor complementarity problems with the implicit $$Z$$-tensors. J. Global Optim. (2022). https:\/\/doi.org\/10.1007\/s10898-022-01263-8","journal-title":"J. Global Optim."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-023-01977-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-023-01977-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-023-01977-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,12]],"date-time":"2023-07-12T13:25:15Z","timestamp":1689168315000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-023-01977-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,17]]},"references-count":26,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["1977"],"URL":"https:\/\/doi.org\/10.1007\/s11590-023-01977-y","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,17]]},"assertion":[{"value":"13 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}