American Institute of Mathematical Sciences

Advanced
2013, 3(3): 583-599. doi: 10.3934/naco.2013.3.583

An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors

Lixing Han 1,

1. 

Department of Mathematics, University of Michigan-Flint, Flint, MI 48502, United States

Received  June 2012 Revised  April 2013 Published  July 2013

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size. $\lambda \in \mathbb{R}$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in \mathbb{R}^n \backslash \{0\}$. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of ${\mathcal A}$. Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.
Keywords: Weakly symmetric tensors, unconstrained optimization., even order, tensor eigenvalues, positive semi-definiteness.
Mathematics Subject Classification: Primary: 65F15, 65K05; Secondary: 15A6.
Citation: Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583
References:
[1]

G. Auchmuty, Unconstrained variational principles for eigenvalues of real symmetric matrices,, SIAM J. Math. Anal., 20 (1989), 1186.  doi: 10.1137/0520078.  Google Scholar

[2]

G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems,, SIAM J. Matrix Anal. Appl., 12 (1991), 690.  doi: 10.1137/0612053.  Google Scholar

[3]

B. W. Bader, T. G. Kolda and others, "MATLAB Tensor Toolbox Version 2.5,", 2012. Available from: \url{http://www.sandia.gov/~tgkolda/TensorToolbox/}., ().   Google Scholar

[4]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor,, Linear Algebra Appl., 438 (2013), 942.  doi: 10.1016/j.laa.2011.05.040.  Google Scholar

[5]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507.   Google Scholar

[6]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalues of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[7]

Y. Dai and C. Hao, A subspace projection method for finding the extreme Z-eigenvalues of supersymmetric positive definite tensor,, A talk given at the International Conference on the Spectral Theory of Tensors, (2012).   Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, Linear Algebra Appl., 438 (2013), 738.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[9]

D. Henrion, J.-B. Lasserre and J. Löfberg, GloptiPoly3: moments, optimization and semidefinite programming,, Optim. Methods Softw., 24 (2009), 761.  doi: 10.1080/10556780802699201.  Google Scholar

[10]

E. Kofidis and Ph. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM J. Matrix Anal. Appl., 23 (2002), 863.  doi: 10.1137/S0895479801387413.  Google Scholar

[11]

T.. Kolda and J.. Mayo, Shifted power method for computing tensor eigenpairs,, SIAM J. Matrix Anal. Appl., 32 (2011), 1095.  doi: 10.1137/100801482.  Google Scholar

[12]

G. Li, L. Qi and G. Yu, "The Z-eigenvalues of a Aymmetric Tensor and Its Application to Spectral Hypergraph Theory,", Department of Applied Mathematics, (2011).   Google Scholar

[13]

L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach,, in, 1 (2005), 129.   Google Scholar

[14]

The Mathworks, Matlab 7.8.0,, 2009., ().   Google Scholar

[15]

J. Nocedal and S. Wright, "Numerical Optimization,", 2nd edition, (2006).   Google Scholar

[16]

A. L. Peressini, F. E. Sullivan and J. J. Uhl, "The Mathematics of Nonlinear Programming,", Springer-Verlag, (1988).   Google Scholar

[17]

L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[18]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[19]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global optimization polynomial optimization problem,, Math. Program., 118 (2009), 301.  doi: 10.1007/s10107-007-0193-6.  Google Scholar

[20]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors,, J. Comput. Appl. Math., 221 (2008), 150.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[21]

L. Qi, G. Yu and E. X. Wu, Higher order positive semi-definite diffusion tensor imaging,, SIAM J. Imaging Sci., 3 (2010), 416.  doi: 10.1137/090755138.  Google Scholar

[22]

L. Qi, G. Yu and Y. Xu, Nonnegative diffusion orientation distribution function,, J. Math. Imaging Vision, 45 (2013), 103.  doi: 10.1007/s10851-012-0346-y.  Google Scholar

show all references

References:
[1]

G. Auchmuty, Unconstrained variational principles for eigenvalues of real symmetric matrices,, SIAM J. Math. Anal., 20 (1989), 1186.  doi: 10.1137/0520078.  Google Scholar

[2]

G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems,, SIAM J. Matrix Anal. Appl., 12 (1991), 690.  doi: 10.1137/0612053.  Google Scholar

[3]

B. W. Bader, T. G. Kolda and others, "MATLAB Tensor Toolbox Version 2.5,", 2012. Available from: \url{http://www.sandia.gov/~tgkolda/TensorToolbox/}., ().   Google Scholar

[4]

D. Cartwright and B. Sturmfels, The number of eigenvalues of a tensor,, Linear Algebra Appl., 438 (2013), 942.  doi: 10.1016/j.laa.2011.05.040.  Google Scholar

[5]

K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507.   Google Scholar

[6]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalues of real symmetric tensors,, J. Math. Anal. Appl., 350 (2009), 416.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[7]

Y. Dai and C. Hao, A subspace projection method for finding the extreme Z-eigenvalues of supersymmetric positive definite tensor,, A talk given at the International Conference on the Spectral Theory of Tensors, (2012).   Google Scholar

[8]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions,, Linear Algebra Appl., 438 (2013), 738.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[9]

D. Henrion, J.-B. Lasserre and J. Löfberg, GloptiPoly3: moments, optimization and semidefinite programming,, Optim. Methods Softw., 24 (2009), 761.  doi: 10.1080/10556780802699201.  Google Scholar

[10]

E. Kofidis and Ph. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors,, SIAM J. Matrix Anal. Appl., 23 (2002), 863.  doi: 10.1137/S0895479801387413.  Google Scholar

[11]

T.. Kolda and J.. Mayo, Shifted power method for computing tensor eigenpairs,, SIAM J. Matrix Anal. Appl., 32 (2011), 1095.  doi: 10.1137/100801482.  Google Scholar

[12]

G. Li, L. Qi and G. Yu, "The Z-eigenvalues of a Aymmetric Tensor and Its Application to Spectral Hypergraph Theory,", Department of Applied Mathematics, (2011).   Google Scholar

[13]

L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach,, in, 1 (2005), 129.   Google Scholar

[14]

The Mathworks, Matlab 7.8.0,, 2009., ().   Google Scholar

[15]

J. Nocedal and S. Wright, "Numerical Optimization,", 2nd edition, (2006).   Google Scholar

[16]

A. L. Peressini, F. E. Sullivan and J. J. Uhl, "The Mathematics of Nonlinear Programming,", Springer-Verlag, (1988).   Google Scholar

[17]

L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[18]

L. Qi, W. Sun and Y. Wang, Numerical multilinear algebra and its applications,, Front. Math. China, 2 (2007), 501.  doi: 10.1007/s11464-007-0031-4.  Google Scholar

[19]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global optimization polynomial optimization problem,, Math. Program., 118 (2009), 301.  doi: 10.1007/s10107-007-0193-6.  Google Scholar

[20]

L. Qi, Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensors,, J. Comput. Appl. Math., 221 (2008), 150.  doi: 10.1016/j.cam.2007.10.012.  Google Scholar

[21]

L. Qi, G. Yu and E. X. Wu, Higher order positive semi-definite diffusion tensor imaging,, SIAM J. Imaging Sci., 3 (2010), 416.  doi: 10.1137/090755138.  Google Scholar

[22]

L. Qi, G. Yu and Y. Xu, Nonnegative diffusion orientation distribution function,, J. Math. Imaging Vision, 45 (2013), 103.  doi: 10.1007/s10851-012-0346-y.  Google Scholar

[1]

Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263
[2]

Yi Xu, Jinjie Liu, Liqun Qi. A new class of positive semi-definite tensors. Journal of Industrial & Management Optimization, 2020, 16 (2) : 933-943. doi: 10.3934/jimo.2018186
[3]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[4]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
[5]

Shenglong Hu, Zheng-Hai Huang, Hong-Yan Ni, Liqun Qi. Positive definiteness of Diffusion Kurtosis Imaging. Inverse Problems & Imaging, 2012, 6 (1) : 57-75. doi: 10.3934/ipi.2012.6.57
[6]

Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411
[7]

Yining Gu, Wei Wu. New bounds for eigenvalues of strictly diagonally dominant tensors. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 203-210. doi: 10.3934/naco.2018012
[8]

Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031
[9]

Yining Gu, Wei Wu. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 775-789. doi: 10.3934/jimo.2018070
[10]

Naoki Chigira, Nobuo Iiyori and Hiroyoshi Yamaki. Nonabelian Sylow subgroups of finite groups of even order. Electronic Research Announcements, 1998, 4: 88-90.

[11]

Chen Ling, Liqun Qi. Some results on $l^k$-eigenvalues of tensor and related spectral radius. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 381-388. doi: 10.3934/naco.2011.1.381
[12]

Steve Rosencrans, Xuefeng Wang, Shan Zhao. Estimating eigenvalues of an anisotropic thermal tensor from transient thermal probe measurements. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5441-5455. doi: 10.3934/dcds.2013.33.5441
[13]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
[14]

Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309
[15]

Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034
[16]

Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595
[17]

Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223
[18]

Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453
[19]

Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465
[20]

Juan Meng, Yisheng Song. Upper bounds for Z$ _1 $-eigenvalues of generalized Hilbert tensors. Journal of Industrial & Management Optimization, 2020, 16 (2) : 911-918. doi: 10.3934/jimo.2018184

 Impact Factor: 

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences