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

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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

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

Received June 2012 Revised April 2013 Published July 2013

Abstract
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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.
[2]	G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems,, SIAM J. Matrix Anal. Appl., 12 (1991), 690. doi: 10.1137/0612053.
[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/}., ().
[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.
[5]	K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507.
[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.
[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).
[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.
[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.
[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.
[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.
[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).
[13]	L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach,, in, 1 (2005), 129.
[14]	The Mathworks, Matlab 7.8.0,, 2009., ().
[15]	J. Nocedal and S. Wright, "Numerical Optimization,", 2nd edition, (2006).
[16]	A. L. Peressini, F. E. Sullivan and J. J. Uhl, "The Mathematics of Nonlinear Programming,", Springer-Verlag, (1988).
[17]	L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.
[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.
[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.

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References:

[1]	G. Auchmuty, Unconstrained variational principles for eigenvalues of real symmetric matrices,, SIAM J. Math. Anal., 20 (1989), 1186. doi: 10.1137/0520078.
[2]	G. Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems,, SIAM J. Matrix Anal. Appl., 12 (1991), 690. doi: 10.1137/0612053.
[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/}., ().
[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.
[5]	K. C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors,, Commun. Math. Sci., 6 (2008), 507.
[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.
[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).
[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.
[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.
[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.
[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.
[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).
[13]	L.-H. Lim, Singular values and eigenvalues of tensors: a variational approach,, in, 1 (2005), 129.
[14]	The Mathworks, Matlab 7.8.0,, 2009., ().
[15]	J. Nocedal and S. Wright, "Numerical Optimization,", 2nd edition, (2006).
[16]	A. L. Peressini, F. E. Sullivan and J. J. Uhl, "The Mathematics of Nonlinear Programming,", Springer-Verlag, (1988).
[17]	L. Qi, Eigenvalues of a real supersymmetric tensor,, J. Symbolic Comput., 40 (2005), 1302. doi: 10.1016/j.jsc.2005.05.007.
[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.
[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.
[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.
[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.
[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.

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