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Completely positive tensors: properties, easily checkable subclasses, and tractable relaxations. (English) Zbl 1349.15026

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
15B48 Positive matrices and their generalizations; cones of matrices
Software:
testmatrix
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References:
[1] P. Alonso, J. Delgado, R. Gallego, and J. M. Pen͂a, Conditioning and accurate computations with Pascal matrices, J. Comput. Appl. Math., 252 (2013), pp. 21–26. · Zbl 1291.65133
[2] N. Arima, S. Kim, and M. Kojima, Extension of completely positive cone relaxation to moment cone relaxation for polynomial optimization, J. Optim. Theory Appl., 168 (2016), pp. 1–17. · Zbl 1336.90068
[3] N. Arima, S. Kim, M. Kojima, and K.-C. Toh, Lagrangian-conic Relaxations, Part II: Applications to Polynomial Optimization Problems, preprint, 2014, .
[4] A. Berman and N. Shaked-Monderer, Completely Positive Matrices, World Scientific, River Edge, NJ, 2003.
[5] R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly, 113 (2006), pp. 221–235. · Zbl 1132.15019
[6] R. Bhatia and H. Kosaki, Mean matrices and infinite divisibility, Linear Algebra Appl., 424 (2007), pp. 36–54. · Zbl 1124.15015
[7] I. M. Bomze, Copositive optimization—recent developments and applications, European J. Oper. Res., 216 (2012), pp. 509–520. · Zbl 1262.90129
[8] R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear Algebra Appl., 174 (1992), pp. 13–23. · Zbl 0755.15012
[9] S. Burer, K. M. Anstreicher, and M. Dür, The difference between \(5×5\) doubly nonnegative and completely positive matrices, Linear Algebra Appl., 431 (2009), pp. 1539–1552. · Zbl 1175.15026
[10] H. Chen, G. Li, and L. Qi, Further results on Cauchy tensors and Hankel tensors, Appl. Math. Comput., 275 (2016), pp. 50–62.
[11] H. Chen, G. Li, and L. Qi, SOS tensor decomposition: Theory and applications, Commun. Math. Sci., 14 (2016), pp. 2073–2100. · Zbl 1351.90148
[12] H. Chen and L. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim., 11 (2015), pp. 1263–1274. · Zbl 1371.15023
[13] Z. Chen and L. Qi, Circulant tensors with applications to spectral hypergraph theory and stochastic process, J. Ind. Manag. Optim., 12 (2016), pp. 1227–1247. · Zbl 1364.15017
[14] A. Cichocki, R. Zdunek, A. H. Phan, and S.-i. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation, John Wiley & Sons, New York, 2009.
[15] W. Ding, L. Qi, and Y. Wei, \(M\)-tensors and nonsingular \(M\)-tensors, Linear Algebra Appl., 439 (2013), pp. 3264–3278.
[16] W. Ding, L. Qi, and Y. Wei, Inheritance properties and sum-of-squares decomposition of Hankel tensors: Theory and algorithms, BIT, 302 (2015), pp. 1–22.
[17] H. Dong, Symmetric tensor approximation hierarchies for the completely positive cone, SIAM J. Optim., 23 (2013), pp. 1850–1866. · Zbl 1291.90129
[18] H. Dong and K. Anstreicher, Separating doubly nonnegative and completely positive matrices, Math. Program., 137 (2013), pp. 131–153. · Zbl 1263.90064
[19] J. Fan and A. Zhou, Completely Positive Tensor Decomposition, preprint arXiv:1411.5149, 2014.
[20] N. J. Higham, The Test Matrix Toolbox for MATLAB (Version \(3.0\)), University of Manchester, Manchester, 1995.
[21] S. Kim, M. Kojima, and K.-C. Toh, A Lagrangian-DNN relaxation: A fast method for computing tight lower bounds for a class of quadratic optimization problems, Math. Program., 156 (2016), pp. 161–187. · Zbl 1342.90123
[22] T. G. Kolda, Numerical optimization for symmetric tensor decomposition, Math. Program., 151 (2015), pp. 225–248. · Zbl 1328.90139
[23] J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796–817. · Zbl 1010.90061
[24] M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry, Springer, New York, 2009, pp. 157–270. · Zbl 1163.13021
[25] C. Li, L. Qi, and Y. Li, \(MB\)-tensors and \(MB_0\)-tensors, Linear Algebra Appl., 484 (2015), pp. 141–153.
[26] Z. Luo, L. Qi, and Y. Ye, Linear operators and positive semidefiniteness of symmetric tensor spaces, Sci. China Math., 58 (2015), pp. 197–212. · Zbl 1308.15025
[27] M. Newman and J. Todd, The evaluation of matrix inversion programs, J. SIAM, 6 (1958), pp. 466–476. · Zbl 0085.34305
[28] J. Pena, J. C. Vera, and L. F. Zuluaga, Completely positive reformulations for polynomial optimization, Math. Program., 151 (2015), pp. 405–431. · Zbl 1328.90114
[29] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302–1324. · Zbl 1125.15014
[30] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), pp. 228–238. · Zbl 1281.15025
[31] L. Qi, \(H^+\)-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci., 12 (2014), pp. 1045–1064.
[32] L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition, Commun. Math. Sci., 13 (2015), pp. 113–125. · Zbl 1331.15020
[33] L. Qi, C. Xu, and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1227–1241. · Zbl 1317.65114
[34] M. Rajesh Kannan, N. Shaked-Monderer, and A. Berman, Some properties of strong \(H\)-tensors and general \(H\)-tensors, Linear Algebra Appl., 476 (2015), pp. 42–55. · Zbl 1316.15029
[35] A. Shashua and T. Hazan, Non-negative tensor factorization with applications to statistics and computer vision, in Proceedings of the 22nd International Conference on Machine Learning, ACM, 2005, pp. 792–799.
[36] Y. Song and L. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl., 451 (2014), pp. 1–14. · Zbl 1292.15027
[37] E. E. Tyrtyshnikov, How bad are Hankel matrices?, Numer. Math., 67 (1994), pp. 261–269. · Zbl 0797.65039
[38] Y. Yang and Q. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2517–2530. · Zbl 1227.15014
[39] A. Yoshise and Y. Matsukawa, On optimization over the doubly nonnegative cone, in Proceedings of the 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), IEEE, 2010, pp. 13–18.
[40] L. Zhang, L. Qi, and G. Zhou, \(M\)-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 437–452.
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