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), 1186-1207. doi: 10.1137/0520078.  Google Scholar

[2]

G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems, SIAM J. Matrix Anal. Appl., 12 (1991), 690-706. 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), 942-952. 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), 507-520.  Google Scholar

[6]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalues of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422. 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, Nankai University, 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), 738-749. 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), 761-779. 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), 863-884. 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), 1095-1124. 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, University of New South Wales, December 2011. Google Scholar

[13]

L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach, in "Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP'05)," 1 (2005), 129-132. Google Scholar

[14]

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

[15]

J. Nocedal and S. Wright, "Numerical Optimization," 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. 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), 501-526. 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), 301-306. 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), 150-157. 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), 416-433. 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), 103-113. 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), 1186-1207. doi: 10.1137/0520078.  Google Scholar

[2]

G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems, SIAM J. Matrix Anal. Appl., 12 (1991), 690-706. 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), 942-952. 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), 507-520.  Google Scholar

[6]

K. C. Chang, K. Pearson and T. Zhang, On eigenvalues of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422. 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, Nankai University, 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), 738-749. 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), 761-779. 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), 863-884. 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), 1095-1124. 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, University of New South Wales, December 2011. Google Scholar

[13]

L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach, in "Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP'05)," 1 (2005), 129-132. Google Scholar

[14]

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

[15]

J. Nocedal and S. Wright, "Numerical Optimization," 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

[16]

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

[17]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. 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), 501-526. 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), 301-306. 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), 150-157. 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), 416-433. 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), 103-113. doi: 10.1007/s10851-012-0346-y.  Google Scholar

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