{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T14:51:03Z","timestamp":1768747863527,"version":"3.49.0"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2012,2,10]],"date-time":"2012-02-10T00:00:00Z","timestamp":1328832000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2012,12]]},"DOI":"10.1007\/s10589-012-9462-2","type":"journal-article","created":{"date-parts":[[2012,2,9]],"date-time":"2012-02-09T16:25:28Z","timestamp":1328804728000},"page":"845-867","source":"Crossref","is-referenced-by-count":8,"title":["On solving biquadratic optimization via semidefinite relaxation"],"prefix":"10.1007","volume":"53","author":[{"given":"Yuning","family":"Yang","sequence":"first","affiliation":[]},{"given":"Qingzhi","family":"Yang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,2,10]]},"reference":[{"key":"9462_CR1","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1137\/S0097539704441629","volume":"35","author":"N. Alon","year":"2006","unstructured":"Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck\u2019s inequality. SIAM J. Comput. 35, 787\u2013803 (2006)","journal-title":"SIAM J. Comput."},{"key":"9462_CR2","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1002\/(SICI)1099-128X(199709\/10)11:5<393::AID-CEM483>3.0.CO;2-L","volume":"11","author":"R. Bro","year":"1997","unstructured":"Bro, R., Jong, S.: A\u00a0fast non-negativity constrained least squares algorithm. J. Chemom. 11, 393\u2013401 (1997)","journal-title":"J. Chemom."},{"key":"9462_CR3","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1002\/(SICI)1099-128X(199807\/08)12:4<223::AID-CEM511>3.0.CO;2-2","volume":"12","author":"R. Bro","year":"1998","unstructured":"Bro, R., Sidiropoulos, N.: Least squares algorithms under unimodality and non-negativity constraints. J. Chemom. 12, 223\u2013247 (1998)","journal-title":"J. Chemom."},{"key":"9462_CR4","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1162\/089976699300016863","volume":"11","author":"J.F. Cardoso","year":"1999","unstructured":"Cardoso, J.F.: High-order contrasts for independent component analysis. Neural Comput. 11, 157\u2013192 (1999)","journal-title":"Neural Comput."},{"key":"9462_CR5","doi-asserted-by":"crossref","DOI":"10.1002\/9780470747278","volume-title":"Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation","author":"A. Cichocki","year":"2009","unstructured":"Cichocki, A., Zdunek, R., Phan, A.H., Amari, S.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. Wiley, New York (2009)"},{"key":"9462_CR6","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0165-1684(94)90029-9","volume":"36","author":"P. Comon","year":"1994","unstructured":"Comon, P.: Independent component analysis, a\u00a0new concept? Signal Process. 36, 287\u2013314 (1994)","journal-title":"Signal Process."},{"key":"9462_CR7","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.laa.2006.08.026","volume":"420","author":"G. Dahl","year":"2007","unstructured":"Dahl, G., Leinaas, J.M., Myrheim, J., Ovrum, E.: A\u00a0tensor product matrix approximation problem in quantum physics. Linear Algebra Appl. 420, 711\u2013725 (2007)","journal-title":"Linear Algebra Appl."},{"key":"9462_CR8","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.1137\/S0895479898346995","volume":"21","author":"L. Lathauwer De","year":"2000","unstructured":"De Lathauwer, L., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(R 1,R 2,\u2026,R N ) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324\u20131342 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9462_CR9","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","volume":"47","author":"A. Einstein","year":"1935","unstructured":"Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777\u2013780 (1935)","journal-title":"Phys. Rev."},{"key":"9462_CR10","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1145\/227683.227684","volume":"42","author":"M.X. Goemans","year":"1995","unstructured":"Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115\u20131145 (1995)","journal-title":"J. ACM"},{"key":"9462_CR11","unstructured":"Grant, M., Boyd, S., CVX: Matlab Software for Disciplined Convex Programming, version\u00a01.2 (2010). http:\/\/cvxr.com\/cvx"},{"key":"9462_CR12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10659-009-9205-5","volume":"97","author":"D. Han","year":"2009","unstructured":"Han, D., Dai, H.H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J.\u00a0Elast. 97, 1\u201313 (2009)","journal-title":"J.\u00a0Elast."},{"key":"9462_CR13","unstructured":"He, S., Li, Z., Zhang, S.: General constrained polynomial optimization: An approximation approach. Technical Report SEEM2009-06, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong (2009)"},{"key":"9462_CR14","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/s10107-010-0409-z","volume":"125","author":"S. He","year":"2010","unstructured":"He, S., Li, Z., Zhang, S.: Approximation algorithms for homogeneous polynomial optimization with quadratic constraints. Math. Program., Ser. B 125, 353\u2013383 (2010)","journal-title":"Math. Program., Ser. B"},{"key":"9462_CR15","doi-asserted-by":"crossref","unstructured":"He, S., Li, Z., Zhang, S.: Approximation algorithms for discrete polynomial optimization. Working paper (2010)","DOI":"10.1007\/s10107-010-0409-z"},{"key":"9462_CR16","doi-asserted-by":"crossref","first-page":"1448","DOI":"10.1137\/070691140","volume":"38","author":"S. Khot","year":"2008","unstructured":"Khot, S., Naor, A.: Linear equations modulo\u00a02 and the L1 diameter of convex bodies. SIAM J. Comput. 38, 1448\u20131463 (2008)","journal-title":"SIAM J. Comput."},{"key":"9462_CR17","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1023\/A:1025794313696","volume":"26","author":"S. Kim","year":"2003","unstructured":"Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26, 143\u2013154 (2003)","journal-title":"Comput. Optim. Appl."},{"key":"9462_CR18","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF00126996","volume":"5","author":"J.K. Knowles","year":"1975","unstructured":"Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of the equations for finite elastostatics for a special material. J.\u00a0Elast. 5, 341\u2013361 (1975)","journal-title":"J.\u00a0Elast."},{"key":"9462_CR19","unstructured":"Kolday, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. (2010)"},{"key":"9462_CR20","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1137\/S1052623400366802","volume":"11","author":"J.B. 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."},{"key":"9462_CR21","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1090\/S0002-9947-01-02898-7","volume":"354","author":"J.B. Lasserre","year":"2001","unstructured":"Lasserre, J.B.: Polynomials nonnegative on a grid and discrete representations. Trans. Am. Math. Soc. 354, 631\u2013649 (2001)","journal-title":"Trans. Am. Math. Soc."},{"key":"9462_CR22","first-page":"129","volume-title":"Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing","author":"L.-H. Lim","year":"2005","unstructured":"Lim, L.-H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, vol.\u00a01, pp. 129\u2013132 (2005)"},{"key":"9462_CR23","doi-asserted-by":"crossref","first-page":"1286","DOI":"10.1137\/080729104","volume":"20","author":"C. Ling","year":"2009","unstructured":"Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20, 1286\u20131310 (2009)","journal-title":"SIAM J. Optim."},{"key":"9462_CR24","author":"C. Ling","year":"2011","unstructured":"Ling, C., Zhang, X., Qi, L.: Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints. Numer. Linear Algebra Appl. (2011). doi: 10.1002\/nla.781","journal-title":"Numer. Linear Algebra Appl."},{"key":"9462_CR25","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1137\/090772952","volume":"20","author":"Z. Luo","year":"2010","unstructured":"Luo, Z., Zhang, S.: A\u00a0semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints. SIAM J. Optim. 20, 1716\u20131736 (2010)","journal-title":"SIAM J. Optim."},{"key":"9462_CR26","first-page":"1","volume":"12","author":"Y. Nesterov","year":"1997","unstructured":"Nesterov, Y.: Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 12, 1\u201320 (1997)","journal-title":"Optim. Methods Softw."},{"key":"9462_CR27","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/S0169-7439(97)00031-2","volume":"38","author":"P. Paatero","year":"1997","unstructured":"Paatero, P.: A\u00a0weighted non-negative least squares algorithm for three-way parafac factor analysis. Chemom. Intell. Lab. Syst. 38, 223\u2013242 (1997)","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"9462_CR28","unstructured":"Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD Dissertation, California Institute of Technology, CA (2000)"},{"key":"9462_CR29","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s10107-003-0387-5","volume":"96","author":"P.A. Parrilo","year":"2003","unstructured":"Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program., Ser. B 96, 293\u2013320 (2003)","journal-title":"Math. Program., Ser. B"},{"key":"9462_CR30","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.: Eigenvalues of a real supersymmetric tensor. J.\u00a0Symb. Comput. 40, 1302\u20131324 (2005)","journal-title":"J.\u00a0Symb. Comput."},{"key":"9462_CR31","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s11464-009-0016-6","volume":"4","author":"L. Qi","year":"2009","unstructured":"Qi, L., Dai, H.H., Han, D.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4, 349\u2013364 (2009)","journal-title":"Front. Math. China"},{"key":"9462_CR32","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s10107-007-0193-6","volume":"118","author":"L. Qi","year":"2009","unstructured":"Qi, L., Wang, F., Wang, Y.: Z-eigenvalue methods for a global polynomial optimization problem. Math. Program., Ser. A 118, 301\u2013316 (2009)","journal-title":"Math. Program., Ser. A"},{"key":"9462_CR33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00377977","volume":"109","author":"P. Rosakis","year":"1990","unstructured":"Rosakis, P.: Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109, 1\u201337 (1990)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9462_CR34","author":"A.M. So","year":"2011","unstructured":"So, A.M.: Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems. Math. Program. (2011). doi: 10.1007\/s10107-011-0464-0","journal-title":"Math. Program."},{"key":"9462_CR35","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF00042193","volume":"44","author":"Y. Wang","year":"1996","unstructured":"Wang, Y., Aron, M.: A\u00a0reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J.\u00a0Elast. 44, 89\u201396 (1996)","journal-title":"J.\u00a0Elast."},{"key":"9462_CR36","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1002\/nla.633","volume":"16","author":"Y. Wang","year":"2009","unstructured":"Wang, Y., Qi, L., Zhang, X.: A\u00a0practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer. Linear Algebra Appl. 16, 589\u2013601 (2009)","journal-title":"Numer. Linear Algebra Appl."},{"key":"9462_CR37","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s101070050006","volume":"87","author":"S. Zhang","year":"2000","unstructured":"Zhang, S.: Quadratic maximization and semidefinite relaxation. Math. Program., Ser. A 87, 453\u2013465 (2000)","journal-title":"Math. Program., Ser. A"},{"key":"9462_CR38","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s10898-010-9545-5","volume":"49","author":"X. Zhang","year":"2010","unstructured":"Zhang, X., Ling, C., Qi, L.: Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. J.\u00a0Glob. Optim. 49, 293\u2013311 (2010)","journal-title":"J.\u00a0Glob. Optim."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9462-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-012-9462-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9462-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T11:10:20Z","timestamp":1640689820000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-012-9462-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,10]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,12]]}},"alternative-id":["9462"],"URL":"https:\/\/doi.org\/10.1007\/s10589-012-9462-2","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,10]]}}}