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doi: 10.3934/jimo.2020147

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 Published  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
[1]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176
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Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
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Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135
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Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
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Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
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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
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Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
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Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
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Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
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Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345
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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
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Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311
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C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020058
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Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
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Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031
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Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
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Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
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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
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Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
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Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

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