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Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities
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 |
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/}., (). 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. |
[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). 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. |
[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). 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).
|
[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. |
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. |
[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/}., (). 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. |
[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). 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. |
[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). 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).
|
[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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