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

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.
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]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[2]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[3]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[4]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001

[5]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

[6]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[7]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[8]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[9]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[10]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[11]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

[12]

Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383

[13]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170

[14]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[15]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[16]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329

[17]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[18]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[19]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[20]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

 Impact Factor: 

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]