# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [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, 2017, 13 (5) : 1-11. 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, 2017, 13 (5) : 1-16. 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, 2017, 13 (5) : 1-8. doi: 10.3934/jimo.2018184

Impact Factor: