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doi: 10.3934/jimo.2020147
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Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems

Ya Li , ShouQiang Du , and  YuanYuan Chen

School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

* Corresponding author: ShouQiang Du

Received  January 2020 Revised  June 2020 Early access September 2020

Tensor eigenvalue complementary problems, as a special class of complementary problems, are the generalization of matrix eigenvalue complementary problems in higher-order. In recent years, tensor eigenvalue complementarity problems have been studied extensively. The research fields of tensor eigenvalue complementarity problems mainly focus on analysis of the theory and algorithms. In this paper, we investigate the solution method for four kinds of tensor eigenvalue complementarity problems with different structures. By utilizing an equivalence relation to unconstrained optimization problems, we propose a modified spectral PRP conjugate gradient method to solve the tensor eigenvalue complementarity problems. Under mild conditions, the global convergence of the given method is also established. Finally, we give related numerical experiments and numerical results compared with inexact Levenberg-Marquardt method, numerical results show the efficiency of the proposed method and also verify our theoretical results.

Keywords: Unconstrained optimization, PRP conjugate gradient method, tensor eigenvalue complementarity problem, global convergence.
Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 15A18.
Citation: Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020147
References:
[1]

S. Adly and H. Rammal, A new method for solving Pareto eigenvalue complementarity problems, Comput. Optim. Appl., 55 (2013), 703-731.  doi: 10.1007/s10589-013-9534-y.  Google Scholar

[2]

M. Al-Baali, Y. Narushima and H. Yabe, A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Comput. Optim. Appl., 60 (2015), 89-110.  doi: 10.1007/s10589-014-9662-z.  Google Scholar

[3]

X. Bai, Z. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, J. Optim. Theory Appl., 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.  Google Scholar

[4]

X. Bai, Z. Huang and X. Li, Stability of solutions and continuity of solution maps of tensor complementarity problems,, Asia-Pacific J. Oper. Res., 36 (2019), 1940002, 19 pp. doi: 10.1142/S0217595919400025.  Google Scholar

[5]

E. Birgin and J. Martinez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim., 43 (2001), 117-128.  doi: 10.1007/s00245-001-0003-0.  Google Scholar

[6]

S. Bojari and M. R. Eslahchi, Two families of scaled three-term conjugate gradient methods with sufficient descent property for nonconvex optimization, Numer. Algorithms, 83 (2020), 901-933.  doi: 10.1007/s11075-019-00709-7.  Google Scholar

[7]

K. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[8]

M. Che, L. Qi and Y. Wei, Positive-defifinite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475-487.  doi: 10.1007/s10957-015-0773-1.  Google Scholar

[9]

Z. Chen, Q. Yang and L. Ye, Generalized eigenvalue complementarity problem for tensors, Pacific J. Optim., 13 (2017), 527-545.   Google Scholar

[10]

Z. Chen and L. Qi, A semismooth Newton method for tensor eigenvalue complementarity problem, Comput. Optim. Appl., 65 (2016), 109-126.  doi: 10.1007/s10589-016-9838-9.  Google Scholar

[11]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[12]

Y. Dai, L. Liao and D. Li, On restart procedures for the conjugate gradient method, Numer. Algor., 35 (2004), 249-260.  doi: 10.1023/B:NUMA.0000021761.10993.6e.  Google Scholar

[13]

W. Ding, L. Qi and Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.  Google Scholar

[14]

S. Du, M. Che and Y. Wei, Stochastic structured tensors to stochastic complementarity problems, Comput. Optim. Appl., 75 (2020), 649-668.  doi: 10.1007/s10589-019-00144-3.  Google Scholar

[15]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 39 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.  Google Scholar

[16]

J. Fan, J. Nie and A. Zhou, Tensor eigenvalue complementarity problems, Math. Program., 170 (2018), 507-539.  doi: 10.1007/s10107-017-1167-y.  Google Scholar

[17]

Z. Huang and L. Qi, Tensor complementarity problems-Part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.  Google Scholar

[18]

Z. Huang and L. Qi, Tensor complementarity problems-Part III: Applications, J. Optim. Theory Appl., 183 (2019), 771-791.  doi: 10.1007/s10957-019-01573-0.  Google Scholar

[19]

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

[20]

H. Li, S. Du, Y. Wang and M. Chen, An inexact Levenberg-Marquardt method for tensor eigenvalue complementarity problem, Pacific J. Optim., 16 (2020), 87-99.   Google Scholar

[21]

C. Ling, H. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Comput. Optim. Appl., 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z.  Google Scholar

[22]

C. Ling, H. He and L. Qi, Higher-degree eigenvalue complementarity problems for tensors, Comput. Optim. Appl., 64 (2016), 149-176.  doi: 10.1007/s10589-015-9805-x.  Google Scholar

[23]

Y. Liu and Q. Yang, A New Eigenvalue Complementarity Problem for Tensor and Matrix, Nankai University, 2018. Google Scholar

[24]

G. Meurant, On prescribing the convergence behavior of the conjugate gradient algorithm, Numer. Algorithms, 84 (2020), 1353-1380.  doi: 10.1007/s11075-019-00851-2.  Google Scholar

[25]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[26]

Q. Ni and L. Qi, A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map, J. Global Optim., 61 (2015), 627-641.  doi: 10.1007/s10898-014-0209-8.  Google Scholar

[27]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[28]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[29]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar

[30]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.  Google Scholar

[31]

L. Qi and Z. Huang, Tensor complementarity problems-Part II: Solution methods, J. Optim. Theory Appl., 183 (2019), 365-385.  doi: 10.1007/s10957-019-01568-x.  Google Scholar

[32]

A. Seeger, Quadratic eigenvalue problems under conic constraints, SIAM J. Matrix Anal. Appl., 32 (2011), 700-721.  doi: 10.1137/100801780.  Google Scholar

[33]

Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308-323.   Google Scholar

[34]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.  doi: 10.1007/s10957-014-0616-5.  Google Scholar

[35]

Y. Song and L. Qi, Tensor complementarity problem and semi-positive tensors, J. Optim. Theory Appl., 169 (2016), 1069-1078.  doi: 10.1007/s10957-015-0800-2.  Google Scholar

[36]

Y. Song and G. Yu, Properties of solution set of tensor complementarity problem, J. Optim. Theory Appl., 170 (2016), 85-96.  doi: 10.1007/s10957-016-0907-0.  Google Scholar

[37]

Z. Wan, Z. Yang and Y. Wang, New spectral PRP conjugate gradient method for unconstrained optimization, Appl. Math. Lett., 24 (2011), 16-22.  doi: 10.1016/j.aml.2010.08.002.  Google Scholar

[38]

X. Wang, M. Che and Y. Wei, Global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors, Numer. Algorithms, 84 (2020), 567-590.  doi: 10.1007/s11075-019-00769-9.  Google Scholar

[39]

Y. Wang, Z. Huang and X. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828.  doi: 10.1080/10556788.2016.1180386.  Google Scholar

[40] Y. Wei and W. Ding, Theory and Computation of Tensors: Multi-Dimensional Arrays, Academic Press, London, 2016.   Google Scholar
[41]

F. Xu and C. Ling, Some properties on Pareto-eigenvalues of higher-order tensors, Oper. Res. Trans., 19 (2015), 34-41.   Google Scholar

[42]

W. Yan and C. Ling, Quadratic eigenvalue complememtarity problem for tensor on second-order cone, Journal of Hangzhou Dianzi University., 38 (2018), 90-93.   Google Scholar

[43] Y. Yang and Q. Yang, A Study on Eigenvalues of Higher-Order Tensors and Related Polynomial Optimization Problems, Science Press, Beijing, 2015.   Google Scholar
[44]

G. Yu, Y. Song, Y. Xu and Z. Yu, Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems, Numer. Algorithms, 80 (2019), 1181-1201.  doi: 10.1007/s11075-018-0522-2.  Google Scholar

[45]

G. Yuan, X. Wang and Z. Sheng, Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions, Numer. Algorithms, 84 (2020), 935-956.  doi: 10.1007/s11075-019-00787-7.  Google Scholar

[46]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.  Google Scholar

[47]

K. Zhang, H. Chen and P. Zhao, A potential reduction method for tensor complementarity problems, J. Ind. Manag. Optim., 15 (2019), 429-443.  doi: 10.3934/jimo.2018049.  Google Scholar

[48]

L. Zhang and W. Zhou, On the global convergence of the Hager-Zhang conjugate gradient method with Armijo line search, Acta Mathematica Scientia., 28 (2008), 840-845.   Google Scholar

[49]

L. Zhang, W. Zhou and D. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numerische Mathematik., 104 (2006), 561–572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[50]

M. Zheng, Y. Zhang and Z. Huang, Global error bounds for the tensor complementarity problem with a P-tensor, J. Ind. Manag. Optim., 15 (2019), 933-946.  doi: 10.3934/jimo.2018078.  Google Scholar

show all references

References:
[1]

S. Adly and H. Rammal, A new method for solving Pareto eigenvalue complementarity problems, Comput. Optim. Appl., 55 (2013), 703-731.  doi: 10.1007/s10589-013-9534-y.  Google Scholar

[2]

M. Al-Baali, Y. Narushima and H. Yabe, A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Comput. Optim. Appl., 60 (2015), 89-110.  doi: 10.1007/s10589-014-9662-z.  Google Scholar

[3]

X. Bai, Z. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, J. Optim. Theory Appl., 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.  Google Scholar

[4]

X. Bai, Z. Huang and X. Li, Stability of solutions and continuity of solution maps of tensor complementarity problems,, Asia-Pacific J. Oper. Res., 36 (2019), 1940002, 19 pp. doi: 10.1142/S0217595919400025.  Google Scholar

[5]

E. Birgin and J. Martinez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim., 43 (2001), 117-128.  doi: 10.1007/s00245-001-0003-0.  Google Scholar

[6]

S. Bojari and M. R. Eslahchi, Two families of scaled three-term conjugate gradient methods with sufficient descent property for nonconvex optimization, Numer. Algorithms, 83 (2020), 901-933.  doi: 10.1007/s11075-019-00709-7.  Google Scholar

[7]

K. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350 (2009), 416-422.  doi: 10.1016/j.jmaa.2008.09.067.  Google Scholar

[8]

M. Che, L. Qi and Y. Wei, Positive-defifinite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475-487.  doi: 10.1007/s10957-015-0773-1.  Google Scholar

[9]

Z. Chen, Q. Yang and L. Ye, Generalized eigenvalue complementarity problem for tensors, Pacific J. Optim., 13 (2017), 527-545.   Google Scholar

[10]

Z. Chen and L. Qi, A semismooth Newton method for tensor eigenvalue complementarity problem, Comput. Optim. Appl., 65 (2016), 109-126.  doi: 10.1007/s10589-016-9838-9.  Google Scholar

[11]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[12]

Y. Dai, L. Liao and D. Li, On restart procedures for the conjugate gradient method, Numer. Algor., 35 (2004), 249-260.  doi: 10.1023/B:NUMA.0000021761.10993.6e.  Google Scholar

[13]

W. Ding, L. Qi and Y. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.  Google Scholar

[14]

S. Du, M. Che and Y. Wei, Stochastic structured tensors to stochastic complementarity problems, Comput. Optim. Appl., 75 (2020), 649-668.  doi: 10.1007/s10589-019-00144-3.  Google Scholar

[15]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 39 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.  Google Scholar

[16]

J. Fan, J. Nie and A. Zhou, Tensor eigenvalue complementarity problems, Math. Program., 170 (2018), 507-539.  doi: 10.1007/s10107-017-1167-y.  Google Scholar

[17]

Z. Huang and L. Qi, Tensor complementarity problems-Part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.  Google Scholar

[18]

Z. Huang and L. Qi, Tensor complementarity problems-Part III: Applications, J. Optim. Theory Appl., 183 (2019), 771-791.  doi: 10.1007/s10957-019-01573-0.  Google Scholar

[19]

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

[20]

H. Li, S. Du, Y. Wang and M. Chen, An inexact Levenberg-Marquardt method for tensor eigenvalue complementarity problem, Pacific J. Optim., 16 (2020), 87-99.   Google Scholar

[21]

C. Ling, H. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Comput. Optim. Appl., 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z.  Google Scholar

[22]

C. Ling, H. He and L. Qi, Higher-degree eigenvalue complementarity problems for tensors, Comput. Optim. Appl., 64 (2016), 149-176.  doi: 10.1007/s10589-015-9805-x.  Google Scholar

[23]

Y. Liu and Q. Yang, A New Eigenvalue Complementarity Problem for Tensor and Matrix, Nankai University, 2018. Google Scholar

[24]

G. Meurant, On prescribing the convergence behavior of the conjugate gradient algorithm, Numer. Algorithms, 84 (2020), 1353-1380.  doi: 10.1007/s11075-019-00851-2.  Google Scholar

[25]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[26]

Q. Ni and L. Qi, A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map, J. Global Optim., 61 (2015), 627-641.  doi: 10.1007/s10898-014-0209-8.  Google Scholar

[27]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[28]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[29]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar

[30]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.  Google Scholar

[31]

L. Qi and Z. Huang, Tensor complementarity problems-Part II: Solution methods, J. Optim. Theory Appl., 183 (2019), 365-385.  doi: 10.1007/s10957-019-01568-x.  Google Scholar

[32]

A. Seeger, Quadratic eigenvalue problems under conic constraints, SIAM J. Matrix Anal. Appl., 32 (2011), 700-721.  doi: 10.1137/100801780.  Google Scholar

[33]

Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308-323.   Google Scholar

[34]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.  doi: 10.1007/s10957-014-0616-5.  Google Scholar

[35]

Y. Song and L. Qi, Tensor complementarity problem and semi-positive tensors, J. Optim. Theory Appl., 169 (2016), 1069-1078.  doi: 10.1007/s10957-015-0800-2.  Google Scholar

[36]

Y. Song and G. Yu, Properties of solution set of tensor complementarity problem, J. Optim. Theory Appl., 170 (2016), 85-96.  doi: 10.1007/s10957-016-0907-0.  Google Scholar

[37]

Z. Wan, Z. Yang and Y. Wang, New spectral PRP conjugate gradient method for unconstrained optimization, Appl. Math. Lett., 24 (2011), 16-22.  doi: 10.1016/j.aml.2010.08.002.  Google Scholar

[38]

X. Wang, M. Che and Y. Wei, Global uniqueness and solvability of tensor complementarity problems for $\mathcal {H}_{+}$-tensors, Numer. Algorithms, 84 (2020), 567-590.  doi: 10.1007/s11075-019-00769-9.  Google Scholar

[39]

Y. Wang, Z. Huang and X. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828.  doi: 10.1080/10556788.2016.1180386.  Google Scholar

[40] Y. Wei and W. Ding, Theory and Computation of Tensors: Multi-Dimensional Arrays, Academic Press, London, 2016.   Google Scholar
[41]

F. Xu and C. Ling, Some properties on Pareto-eigenvalues of higher-order tensors, Oper. Res. Trans., 19 (2015), 34-41.   Google Scholar

[42]

W. Yan and C. Ling, Quadratic eigenvalue complememtarity problem for tensor on second-order cone, Journal of Hangzhou Dianzi University., 38 (2018), 90-93.   Google Scholar

[43] Y. Yang and Q. Yang, A Study on Eigenvalues of Higher-Order Tensors and Related Polynomial Optimization Problems, Science Press, Beijing, 2015.   Google Scholar
[44]

G. Yu, Y. Song, Y. Xu and Z. Yu, Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems, Numer. Algorithms, 80 (2019), 1181-1201.  doi: 10.1007/s11075-018-0522-2.  Google Scholar

[45]

G. Yuan, X. Wang and Z. Sheng, Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions, Numer. Algorithms, 84 (2020), 935-956.  doi: 10.1007/s11075-019-00787-7.  Google Scholar

[46]

L. Zhang, L. Qi and G. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452.  doi: 10.1137/130915339.  Google Scholar

[47]

K. Zhang, H. Chen and P. Zhao, A potential reduction method for tensor complementarity problems, J. Ind. Manag. Optim., 15 (2019), 429-443.  doi: 10.3934/jimo.2018049.  Google Scholar

[48]

L. Zhang and W. Zhou, On the global convergence of the Hager-Zhang conjugate gradient method with Armijo line search, Acta Mathematica Scientia., 28 (2008), 840-845.   Google Scholar

[49]

L. Zhang, W. Zhou and D. Li, Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numerische Mathematik., 104 (2006), 561–572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[50]

M. Zheng, Y. Zhang and Z. Huang, Global error bounds for the tensor complementarity problem with a P-tensor, J. Ind. Manag. Optim., 15 (2019), 933-946.  doi: 10.3934/jimo.2018078.  Google Scholar

Figure 1.  Numerical results of Example 4.1 with random initial points
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Figure 2.  Numerical results of Example 4.5 with different initial points
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Table 1.  The numerical results of Example 4.1
Eigvalue Eigvector No
1.9406 $ (0.4982, 0.5018)^T $ 24
2.0469 $ (0.8817, 0.1183)^T $ 5
2.6308 $ (0.0000, 1.0000)^T $ 1
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Eigvalue Eigvector No
1.9406 $ (0.4982, 0.5018)^T $ 24
2.0469 $ (0.8817, 0.1183)^T $ 5
2.6308 $ (0.0000, 1.0000)^T $ 1
Table 2.  The numerical results of Example 4.2
Alg. Eigvalue Eigvector No. K
Algorithm 3.1 0.3024 $ (0.5479, 1.0000, 0.4521)^T $ 3 69
Algorithm 3.1 0.2356 $ (0.5037, 0.4963, 0.0000)^T $ 2 56
Algorithm 3.1 0.2089 $ (0.0000, 0.4911, 0.5089)^T $ 1 41
Algorithm 3.1 0.0771 $ (0.3328, 0.3372, 0.3301)^T $ 3 183
Algorithm 3.1 $ failure $ $ - $ 1 -
ILMM 0.2089 $ (0.0000, 0.4911, 0.5089)^T $ 2 26
ILMM 0.2356 $ (0.5037, 0.4963, 0.0000)^T $ 2 41
ILMM 0.3024 $ (0.5479, 1.0000, 0.4521)^T $ 1 45
ILMM 0.9807 $ (0.0000, 1.0000, 0.0000)^T $ 1 119
ILMM $ failure $ - 4 -
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Alg. Eigvalue Eigvector No. K
Algorithm 3.1 0.3024 $ (0.5479, 1.0000, 0.4521)^T $ 3 69
Algorithm 3.1 0.2356 $ (0.5037, 0.4963, 0.0000)^T $ 2 56
Algorithm 3.1 0.2089 $ (0.0000, 0.4911, 0.5089)^T $ 1 41
Algorithm 3.1 0.0771 $ (0.3328, 0.3372, 0.3301)^T $ 3 183
Algorithm 3.1 $ failure $ $ - $ 1 -
ILMM 0.2089 $ (0.0000, 0.4911, 0.5089)^T $ 2 26
ILMM 0.2356 $ (0.5037, 0.4963, 0.0000)^T $ 2 41
ILMM 0.3024 $ (0.5479, 1.0000, 0.4521)^T $ 1 45
ILMM 0.9807 $ (0.0000, 1.0000, 0.0000)^T $ 1 119
ILMM $ failure $ - 4 -
Table 3.  The numerical results of Example 4.3
Eigvalue Eigvector No
0.8867 $ (1.0000, 0.0000)^T $ 2
0.9533 $ (0.0000, 1.0000)^T $ 4
0.6 $ (0.6, 0.4)^T $ 14
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Eigvalue Eigvector No
0.8867 $ (1.0000, 0.0000)^T $ 2
0.9533 $ (0.0000, 1.0000)^T $ 4
0.6 $ (0.6, 0.4)^T $ 14
Table 4.  The numerical results of Example 4.4
Eigvalue Eigvector No
2.9167 $ (0.0232, 0.0000, 0.9967)^T $ 3
2.9865 $ (0.0712, 0.0252, 0.9971)^T $ 2
0.3 $ (0.1, 0, 0.1)^T $ 1
failure 4
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Eigvalue Eigvector No
2.9167 $ (0.0232, 0.0000, 0.9967)^T $ 3
2.9865 $ (0.0712, 0.0252, 0.9971)^T $ 2
0.3 $ (0.1, 0, 0.1)^T $ 1
failure 4
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