{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T18:11:55Z","timestamp":1785262315206,"version":"3.55.0"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2017,8,16]],"date-time":"2017-08-16T00:00:00Z","timestamp":1502841600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2018,1]]},"DOI":"10.1007\/s10589-017-9938-1","type":"journal-article","created":{"date-parts":[[2017,8,16]],"date-time":"2017-08-16T07:20:05Z","timestamp":1502868005000},"page":"133-158","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["Copositive tensor detection and its applications in physics and hypergraphs"],"prefix":"10.1007","volume":"69","author":[{"given":"Haibin","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zheng-Hai","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liqun","family":"Qi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2017,8,16]]},"reference":[{"key":"9938_CR1","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s10957-016-0903-4","volume":"170","author":"XL Bai","year":"2016","unstructured":"Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72\u201384 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"7","key":"9938_CR2","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1016\/j.laa.2007.09.035","volume":"428","author":"S Bundfuss","year":"2008","unstructured":"Bundfuss, S., D\u00fcr, M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428(7), 1511\u20131523 (2008)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR3","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":"9938_CR4","unstructured":"Chen, H., Chen, Y., Li, G., Qi, L.: Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs (2016). \n                        arXiv:1511.02328v2"},{"key":"9938_CR5","doi-asserted-by":"publisher","unstructured":"Chen, H., Huang, Z.H., Qi, L.: Copositivity detection of tensors: theory and algorithm. J. Optim. Theory Appl. (2017). doi:\n                        10.1007\/s10957-017-1131-2","DOI":"10.1007\/s10957-017-1131-2"},{"key":"9938_CR6","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.4310\/CMS.2016.v14.n8.a1","volume":"8","author":"H Chen","year":"2016","unstructured":"Chen, H., Li, G., Qi, L.: SOS tensor decomposition: theory and applications. Commun. Math. Sci. 8, 2073\u20132100 (2016)","journal-title":"Commun. Math. Sci."},{"key":"9938_CR7","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.3934\/jimo.2015.11.1263","volume":"11","author":"H Chen","year":"2015","unstructured":"Chen, H., Qi, L.: Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J. Ind. Manag. Optim. 11, 1263\u20131274 (2015)","journal-title":"J. Ind. Manag. Optim."},{"key":"9938_CR8","doi-asserted-by":"crossref","first-page":"3268","DOI":"10.1016\/j.laa.2011.11.018","volume":"436","author":"J Cooper","year":"2012","unstructured":"Cooper, J., Dutle, A.: Spectra of uniform hypergraphs. Linear Algebra Appl. 436, 3268\u20133292 (2012)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR9","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s10589-013-9594-z","volume":"57","author":"PJC Dickinson","year":"2014","unstructured":"Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57, 403\u2013415 (2014)","journal-title":"Comput. Optim. Appl."},{"key":"9938_CR10","unstructured":"Ding, W., Luo, Z., Qi, L.: P-tensors, \n                        $$P _0 $$\n                        \n                            \n                                \n                                    P\n                                    0\n                                \n                            \n                        \n                    -tensors, and tensor complementarity problem (2015). \n                        arXiv:1507.06731"},{"key":"9938_CR11","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1145\/2512329","volume":"60","author":"CJ Hillar","year":"2013","unstructured":"Hillar, C.J., Lim, L.H.: Most tensor problems are NP-hard. J. ACM (JACM) 60, 45 (2013)","journal-title":"J. ACM (JACM)"},{"key":"9938_CR12","doi-asserted-by":"crossref","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, 557\u2013576 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"9938_CR13","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.laa.2015.02.034","volume":"476","author":"MR Kannan","year":"2015","unstructured":"Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong \n                        $$H$$\n                        \n                            \n                                H\n                            \n                        \n                    -tensors and general \n                        $$H$$\n                        \n                            \n                                H\n                            \n                        \n                    -tensors. Linear Algebra Appl. 476, 42\u201355 (2015)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR14","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1140\/epjc\/s10052-016-4160-3","volume":"76","author":"K Kannike","year":"2016","unstructured":"Kannike, K.: Vacuum stability of a general scalar potential of a few fields. Eur. Phys. J. C 76, 324 (2016)","journal-title":"Eur. Phys. J. C"},{"key":"9938_CR15","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s10107-015-0895-0","volume":"151","author":"T Kolda","year":"2015","unstructured":"Kolda, T.: Numerical optimization for symmetric tensor decomposition. Math. Program. Ser. B 151, 225\u2013248 (2015)","journal-title":"Math. Program. Ser. B"},{"key":"9938_CR16","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.laa.2014.10.027","volume":"466","author":"C Li","year":"2015","unstructured":"Li, C., Li, Y.: Double B tensors and quasi-double B tensors. Linear Algebra Appl. 466, 343\u2013356 (2015)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR17","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.laa.2015.06.030","volume":"484","author":"C Li","year":"2015","unstructured":"Li, C., Qi, L., Li, Y.: MB-tensors and MB\n                        $$_0$$\n                        \n                            \n                                \n                                    \n                                    0\n                                \n                            \n                        \n                    -tensors. Linear Algebra Appl. 484, 141\u2013153 (2015)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2013.04.022","volume":"255","author":"C Li","year":"2014","unstructured":"Li, C., Wang, F., Zhao, J., Zhu, Y., Li, Y.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1\u201314 (2014)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"9938_CR19","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s10589-015-9767-z","volume":"63","author":"C Ling","year":"2016","unstructured":"Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63(1), 143\u2013168 (2016)","journal-title":"Comput. Optim. Appl."},{"key":"9938_CR20","unstructured":"L\u00f6fberg, J.: YALMIP : a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, (2004)"},{"key":"9938_CR21","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1109\/TAC.2009.2017144","volume":"54","author":"J L\u00f6fberg","year":"2009","unstructured":"L\u00f6fberg, J.: Pre- and post-processing sums-of-squares programs in practice. IEEE Trans. Autom. Control 54, 1007\u20131011 (2009)","journal-title":"IEEE Trans. Autom. Control"},{"key":"9938_CR22","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1137\/15M1025220","volume":"37","author":"Z Luo","year":"2016","unstructured":"Luo, Z., Qi, L.: Completely positive tensors: properties, easily checkable subclasses and tractable relaxations. SIAM J. Matrix Anal. Appl. 37, 1675\u20131698 (2016)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"9938_CR23","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF02592948","volume":"39","author":"KG Murty","year":"1987","unstructured":"Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117\u2013129 (1987)","journal-title":"Math. Program."},{"issue":"2","key":"9938_CR24","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s10107-014-0822-9","volume":"151","author":"J Pena","year":"2014","unstructured":"Pena, J., Vera, J.C., Zuluaga, L.F.: Completely positive reformulations for polynomial optimization. Math. Program. 151(2), 405\u2013431 (2014)","journal-title":"Math. Program."},{"key":"9938_CR25","doi-asserted-by":"crossref","first-page":"1302","DOI":"10.1016\/j.jsc.2005.05.007","volume":"40","author":"L Qi","year":"2005","unstructured":"Qi, L.: Eigenvalue of a real supersymmetric tensor. J. Symb. Comput. 40, 1302\u20131324 (2005)","journal-title":"J. Symb. Comput."},{"key":"9938_CR26","doi-asserted-by":"crossref","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":"9938_CR27","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.4310\/CMS.2014.v12.n6.a3","volume":"12","author":"L Qi","year":"2014","unstructured":"Qi, L.: H\n                        $$^+$$\n                        \n                            \n                                \n                                    \n                                    +\n                                \n                            \n                        \n                    -eigenvalues of Laplacian and signless Laplacian tensors. Commun. Math. Sci. 12, 1045\u20131064 (2014)","journal-title":"Commun. Math. Sci."},{"key":"9938_CR28","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.laa.2014.05.026","volume":"457","author":"L Qi","year":"2014","unstructured":"Qi, L., Song, Y.: An even order symmetric B tensor is positive definite. Linear Algebra Appl. 457, 303\u2013312 (2014)","journal-title":"Linear Algebra Appl."},{"issue":"4","key":"9938_CR29","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1137\/13092232X","volume":"35","author":"L Qi","year":"2014","unstructured":"Qi, L., Xu, C., Xu, Y.: Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J. Matrix Anal. Appl. 35(4), 1227\u20131241 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9938_CR30","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1016\/j.laa.2013.07.010","volume":"439","author":"JY Shao","year":"2013","unstructured":"Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350\u20132366 (2013)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"9938_CR31","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1080\/03081087.2013.851198","volume":"63","author":"Y Song","year":"2015","unstructured":"Song, Y., Qi, L.: Necessary and sufficient conditions for copositive tensors. Linear and Multilinear Algebra 63(1), 120\u2013131 (2015)","journal-title":"Linear and Multilinear Algebra"},{"key":"9938_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-016-1104-7","author":"Y Song","year":"2016","unstructured":"Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. (2016). doi:\n                        10.1007\/s11590-016-1104-7","journal-title":"Optim. Lett."},{"key":"9938_CR33","doi-asserted-by":"crossref","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":"9938_CR34","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s10898-015-0343-y","volume":"64","author":"Y Song","year":"2016","unstructured":"Song, Y., Qi, L.: Eigenvalue analysis of constrained minimization problem for homogeneous polynomials. J. Global Optim. 64, 563\u2013575 (2016)","journal-title":"J. Global Optim."},{"issue":"3","key":"9938_CR35","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s10898-011-9766-2","volume":"52","author":"J Sponsel","year":"2012","unstructured":"Sponsel, J., Bundfuss, S., D\u00fcr, M.: An improved algorithm to test copositivity. J. Global Optim. 52(3), 537\u2013551 (2012)","journal-title":"J. Global Optim."},{"issue":"12","key":"9938_CR36","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1080\/10556789908805766","volume":"11","author":"JF Sturm","year":"1999","unstructured":"Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(12), 625\u2013653 (1999)","journal-title":"Optim. Methods Softw."},{"key":"9938_CR37","unstructured":"Tanaka, A., Yoshise, A.: Tractable subcones and LP-based algorithms for testing copositivity (2016). \n                        arXiv:1601.06878"},{"issue":"4","key":"9938_CR38","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1080\/10556788.2016.1180386","volume":"31","author":"Y Wang","year":"2016","unstructured":"Wang, Y., Huang, Z.H., Bai, X.L.: Exceptionally regular tensors and complementarity problems. Optim. Methods Softw. 31(4), 815\u2013828 (2016)","journal-title":"Optim. Methods Softw."},{"issue":"11","key":"9938_CR39","doi-asserted-by":"crossref","first-page":"2784","DOI":"10.1016\/j.laa.2011.04.038","volume":"435","author":"J Xu","year":"2011","unstructured":"Xu, J., Yao, Y.: An algorithm for determining copositive matrices. Linear Algebra Appl. 435(11), 2784\u20132792 (2011)","journal-title":"Linear Algebra Appl."},{"key":"9938_CR40","doi-asserted-by":"crossref","unstructured":"Yoshise, A., Matsukawa, Y.: On optimization over the doubly nonnegative cone. In: 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), IEEE pp. 13\u201318 (2010)","DOI":"10.1109\/CACSD.2010.5612811"},{"key":"9938_CR41","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1137\/130915339","volume":"35","author":"L Zhang","year":"2014","unstructured":"Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35, 437\u2013452 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-017-9938-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9938-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9938-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,1,3]],"date-time":"2018-01-03T17:59:28Z","timestamp":1515002368000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-017-9938-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,16]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["9938"],"URL":"https:\/\/doi.org\/10.1007\/s10589-017-9938-1","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,16]]}}}