# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022035
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## Pareto eigenvalue inclusion intervals for tensors

 School of Mathematics, Tianjin University, Tianjin 300350, China

*Corresponding author: Zheng-Hai Huang

Received  August 2021 Revised  December 2021 Early access March 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos 11871051 and 12171357)

A Pareto eigenvalue of a tensor
 ${\mathcal A}$
of order
 $m$
and dimension
 $n$
is a real number
 $\lambda$
for which the complementarity problem
 $\mathbf{0}\leq {\mathbf x} \bot (\lambda{\mathcal E}{\mathbf x}^{m-1}- {\mathcal A}{\mathbf x}^{m-1}) \geq \mathbf{0}$
 ${\mathbf x}\in \mathbb{R}^n$
, where
 ${\mathcal E}$
is an identity tensor. In this paper, we investigate some basic properties of Pareto eigenvalues, including an equivalent condition for the existence of strict Pareto eigenvalues and the nonnegative conditions of Pareto eigenvalues. Then we focus on the estimation of the bounds of Pareto eigenvalues. Specifically, we propose several Pareto eigenvalue inclusion intervals, and discuss the relationships among them and the known result, which demonstrate that the inclusion intervals obtained are tighter than the known one. Finally, as an application of an achieved inclusion intervals, we propose a sufficient condition for judging that a tensor is strictly copositive.
Citation: Yang Xu, Zheng-Hai Huang. Pareto eigenvalue inclusion intervals for tensors. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022035
##### References:
 [1] S. Adly and H. Rammal, A new method for solving Pareto eigenvalue complementarity problems, Computational Optimization and Applications, 55 (2013), 703-731.  doi: 10.1007/s10589-013-9534-y. [2] A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, New York, 2003. [3] J.B. Baillon and A. Seeger, New results on Pareto spectra, Linear Algebra and its Applications, 588 (2020), 338-363.  doi: 10.1016/j.laa.2019.11.029. [4] C.P. Brás, A. Fischer, J.J. Júdice, K. Sch${\rm\ddot{o}}$enefeld and S. Seifert, A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem, Applied Mathematics and Computation, 294 (2017), 36-48.  doi: 10.1016/j.amc.2016.09.005. [5] A. Brauer, Limits for the characteristic roots of a matrix II, Duke Mathematical Journal, 14 (1947), 21-26.  doi: 10.1215/S0012-7094-47-01403-8. [6] Z. Chen and L. Qi, A semismooth Newton method for tensor eigenvalue complementarity problem, Computational Optimization and Applications, 65 (2016), 109-126.  doi: 10.1007/s10589-016-9838-9. [7] Z. Chen, Q. Yang and L. Ye, Generalized eigenvalue complementarity problem for tensors, Pacific Journal of Optimization, 13 (2017), 527-545. [8] L. Cheng, X. Zhang and G. Ni, A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems, Journal of Global Optimization, 79 (2021), 715-732.  doi: 10.1007/s10898-020-00954-4. [9] A.P. da Costa, I.N. Figueiredo, J.J. Júdice and J.A.C. Martins, A complementarity eigenproblem in the stability analysis of finite dimensional elastic systems with frictional contact, in Complementarity: Applications, Algorithms and Extensions (eds. M.C. Ferris, O.L. Mangasarian and J.-S. Pang), Springer, (2001), 67–83. doi: 10.1007/978-1-4757-3279-5_4. [10] J. Fan, J. Nie and R. Zhao, The maximum tensor complementarity eigenvalues, Optimization Methods & Software, 35 (2020), 1179-1190.  doi: 10.1080/10556788.2018.1528251. [11] J. Fan, J. Nie and A. Zhou, Tensor eigenvalue complementarity problems, Mathematical Programming, 170 (2018), 507-539.  doi: 10.1007/s10107-017-1167-y. [12] L.M. Fernandes, J.J. Júdice, H. D., Sh erali and M.A. Forjaz, On an enumerative algorithm for solving eigenvalue complementarity problems, Computational Optimization and Applications, 59 (2014), 113-134.  doi: 10.1007/s10589-012-9529-0. [13] L.M. Fernandes, J.J. Júdice, H.D. Sherali and M. Fukushima, On the computation of all eigenvalues for the eigenvalue complementarity problem, Journal of Global Optimization, 59 (2014), 307-326.  doi: 10.1007/s10898-014-0165-3. [14] S.A. Gersgorin, Uber die abgrenzung der eigenwerte einer matrix, Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na, 6 (1931), 749-754. [15] J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106-137.  doi: 10.1016/j.aml.2019.106137. [16] W. Hu, L. Lu, C. Yin and G. Yu, A smoothing Newton method for tensor eigenvalue complementarity problems, Pacific Journal of Optimization, 13 (2017), 243-253. [17] Z.-H. Huang, X. Li and Y. Wang, Bi-block positive semidefiniteness of bi-block symmetric tensors, Frontiers of Mathematics in China, 16 (2021), 141-169.  doi: 10.1007/s11464-021-0874-0. [18] J.J. Júdice, H.D. Sherali and I.M. Ribeiro, The eigenvalue complementarity problem, Computational Optimization and Applications, 37 (2007), 139-156.  doi: 10.1007/s10589-007-9017-0. [19] C. Li, Z. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra and its Applications, 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023. [20] H. Li, S. Du, Y. Wang and M. Chen, An inexact Levenberg-Marquardt method for tensor eigenvalue complementarity problem, Pacific Journal of Optimization, 16 (2020), 87-99. [21] C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors, Linear and Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582. [22] C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858. [23] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013. [24] C. Ling, H. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Computational Optimization and Applications, 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z. [25] C. Ling, H. He and L. Qi, Higher-degree eigenvalue complementarity problems for tensors, Computational Optimization and Applications, 64 (2016), 149-176.  doi: 10.1007/s10589-015-9805-x. [26] L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007. [27] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra and its Applications, 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015. [28] M. Queiroz, J.J. Júdice and C. Humes, The symmetric eigenvalue complementarity problem, Mathematics of Computation, 73 (2004), 1849-1863.  doi: 10.1090/S0025-5718-03-01614-4. [29] A. Seeger, Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions, Linear Algebra and its Applications, 292 (1999), 1-14.  doi: 10.1016/S0024-3795(99)00004-X. [30] Y. Song and L. Qi, Eigenvalue analysis of constrained minimization problem for homogeneous polynomial, Journal of Global Optimization, 64 (2016), 563-575.  doi: 10.1007/s10898-015-0343-y. [31] Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Annals of Applied Mathematics, 33 (2017), 308-323. [32] W. Tong, H. He, C. Ling and L. Qi, A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems, Numerical Algebra, Control and Optimization, 10 (2020), 425-437.  doi: 10.3934/naco.2020042. [33] G. Yu, Y. Song, Y. Xu and Z. Yu, Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems, Numerical Algorithms, 80 (2019), 1181-1201.  doi: 10.1007/s11075-018-0522-2. [34] L. Zhang and C. Chen, A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications, Mathematics of Computation, 90 (2021), 215-231.  doi: 10.1090/mcom/3558. [35] R. Zhao and J. Fan, Higher-degree tensor eigenvalue complementarity problems, Computational Optimization and Applications, 75 (2020), 799-816.  doi: 10.1007/s10589-019-00159-w.

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##### References:
 [1] S. Adly and H. Rammal, A new method for solving Pareto eigenvalue complementarity problems, Computational Optimization and Applications, 55 (2013), 703-731.  doi: 10.1007/s10589-013-9534-y. [2] A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, New York, 2003. [3] J.B. Baillon and A. Seeger, New results on Pareto spectra, Linear Algebra and its Applications, 588 (2020), 338-363.  doi: 10.1016/j.laa.2019.11.029. [4] C.P. Brás, A. Fischer, J.J. Júdice, K. Sch${\rm\ddot{o}}$enefeld and S. Seifert, A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem, Applied Mathematics and Computation, 294 (2017), 36-48.  doi: 10.1016/j.amc.2016.09.005. [5] A. Brauer, Limits for the characteristic roots of a matrix II, Duke Mathematical Journal, 14 (1947), 21-26.  doi: 10.1215/S0012-7094-47-01403-8. [6] Z. Chen and L. Qi, A semismooth Newton method for tensor eigenvalue complementarity problem, Computational Optimization and Applications, 65 (2016), 109-126.  doi: 10.1007/s10589-016-9838-9. [7] Z. Chen, Q. Yang and L. Ye, Generalized eigenvalue complementarity problem for tensors, Pacific Journal of Optimization, 13 (2017), 527-545. [8] L. Cheng, X. Zhang and G. Ni, A semidefinite relaxation method for second-order cone tensor eigenvalue complementarity problems, Journal of Global Optimization, 79 (2021), 715-732.  doi: 10.1007/s10898-020-00954-4. [9] A.P. da Costa, I.N. Figueiredo, J.J. Júdice and J.A.C. Martins, A complementarity eigenproblem in the stability analysis of finite dimensional elastic systems with frictional contact, in Complementarity: Applications, Algorithms and Extensions (eds. M.C. Ferris, O.L. Mangasarian and J.-S. Pang), Springer, (2001), 67–83. doi: 10.1007/978-1-4757-3279-5_4. [10] J. Fan, J. Nie and R. Zhao, The maximum tensor complementarity eigenvalues, Optimization Methods & Software, 35 (2020), 1179-1190.  doi: 10.1080/10556788.2018.1528251. [11] J. Fan, J. Nie and A. Zhou, Tensor eigenvalue complementarity problems, Mathematical Programming, 170 (2018), 507-539.  doi: 10.1007/s10107-017-1167-y. [12] L.M. Fernandes, J.J. Júdice, H. D., Sh erali and M.A. Forjaz, On an enumerative algorithm for solving eigenvalue complementarity problems, Computational Optimization and Applications, 59 (2014), 113-134.  doi: 10.1007/s10589-012-9529-0. [13] L.M. Fernandes, J.J. Júdice, H.D. Sherali and M. Fukushima, On the computation of all eigenvalues for the eigenvalue complementarity problem, Journal of Global Optimization, 59 (2014), 307-326.  doi: 10.1007/s10898-014-0165-3. [14] S.A. Gersgorin, Uber die abgrenzung der eigenwerte einer matrix, Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na, 6 (1931), 749-754. [15] J. He, C. Li and Y. Wei, M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity, Applied Mathematics Letters, 102 (2020), 106-137.  doi: 10.1016/j.aml.2019.106137. [16] W. Hu, L. Lu, C. Yin and G. Yu, A smoothing Newton method for tensor eigenvalue complementarity problems, Pacific Journal of Optimization, 13 (2017), 243-253. [17] Z.-H. Huang, X. Li and Y. Wang, Bi-block positive semidefiniteness of bi-block symmetric tensors, Frontiers of Mathematics in China, 16 (2021), 141-169.  doi: 10.1007/s11464-021-0874-0. [18] J.J. Júdice, H.D. Sherali and I.M. Ribeiro, The eigenvalue complementarity problem, Computational Optimization and Applications, 37 (2007), 139-156.  doi: 10.1007/s10589-007-9017-0. [19] C. Li, Z. Chen and Y. Li, A new eigenvalue inclusion set for tensors and its applications, Linear Algebra and its Applications, 481 (2015), 36-53.  doi: 10.1016/j.laa.2015.04.023. [20] H. Li, S. Du, Y. Wang and M. Chen, An inexact Levenberg-Marquardt method for tensor eigenvalue complementarity problem, Pacific Journal of Optimization, 16 (2020), 87-99. [21] C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors, Linear and Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582. [22] C. Li, Y. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858. [23] S. Li, C. Li and Y. Li, M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, Journal of Computational and Applied Mathematics, 356 (2019), 391-401.  doi: 10.1016/j.cam.2019.01.013. [24] C. Ling, H. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Computational Optimization and Applications, 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z. [25] C. Ling, H. He and L. Qi, Higher-degree eigenvalue complementarity problems for tensors, Computational Optimization and Applications, 64 (2016), 149-176.  doi: 10.1007/s10589-015-9805-x. [26] L. Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007. [27] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra and its Applications, 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015. [28] M. Queiroz, J.J. Júdice and C. Humes, The symmetric eigenvalue complementarity problem, Mathematics of Computation, 73 (2004), 1849-1863.  doi: 10.1090/S0025-5718-03-01614-4. [29] A. Seeger, Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions, Linear Algebra and its Applications, 292 (1999), 1-14.  doi: 10.1016/S0024-3795(99)00004-X. [30] Y. Song and L. Qi, Eigenvalue analysis of constrained minimization problem for homogeneous polynomial, Journal of Global Optimization, 64 (2016), 563-575.  doi: 10.1007/s10898-015-0343-y. [31] Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Annals of Applied Mathematics, 33 (2017), 308-323. [32] W. Tong, H. He, C. Ling and L. Qi, A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems, Numerical Algebra, Control and Optimization, 10 (2020), 425-437.  doi: 10.3934/naco.2020042. [33] G. Yu, Y. Song, Y. Xu and Z. Yu, Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems, Numerical Algorithms, 80 (2019), 1181-1201.  doi: 10.1007/s11075-018-0522-2. [34] L. Zhang and C. Chen, A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications, Mathematics of Computation, 90 (2021), 215-231.  doi: 10.1090/mcom/3558. [35] R. Zhao and J. Fan, Higher-degree tensor eigenvalue complementarity problems, Computational Optimization and Applications, 75 (2020), 799-816.  doi: 10.1007/s10589-019-00159-w.
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