# American Institute of Mathematical Sciences

April  2019, 15(2): 429-443. doi: 10.3934/jimo.2018049

## A potential reduction method for tensor complementarity problems

 1 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2 School of Management Science, Qufu Normal University, Rizhao, Shandong 276800, China 3 College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China

* Corresponding author: Haibin Chen, chenhaibin508@163.com

Received  June 2017 Revised  December 2017 Published  April 2018

Fund Project: The authors' work are supported by the Natural Science Foundation of China (Grant No. 11601261, 11671228, 11771003), the Natural Science Foundation of Shandong Province (Grant No. ZR2016AQ12), and the China Postdoctoral Science Foundation (Grant No. 2017M622163)

As an extension of linear complementary problem, tensor complementary problem has been effectively applied in $n$-person noncooperative game. And a multitude of researchers have focused on its properties and theories, while the valid algorithms for tensor complementary problem is still deficient. In this paper, stimulated by the potential reduction method for linear complementarity problem, we present a new algorithm for the tensor complementarity problem, which combines the idea of damped Newton method and the interior point method. Utilizing the new algorithm, we settle the tensor complementary problem with the underlying tensor being diagonalizable and positive definite. Furthermore, the global convergence of the iterative scheme is theoretically guaranteed and the given preliminary numerical experiments indicate the efficiency of the method.

Citation: Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049
##### References:
 [1] X. L. Bai, Z. H. 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. [2] L. Castello and H. Clercx, Geometrical statistics of the vorticity vector and the strain rate tensor in rotating turbulence, J. Turbul., 14 (2013), 19-36. doi: 10.1080/14685248.2013.866241. [3] M. L. Che, L. Q. Qi and Y. M. Wei, Positive-definite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475-487. doi: 10.1007/s10957-015-0773-1. [4] H. B. Chen, Y. N. Chen, G. Y. Li and L. Q. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Lin. Alg. Appl., 25 (2018), e2125. [5] H. B. Chen, Z. H. Huang and L. Q. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optim. Theory Appl., 174 (2017), 746-761. doi: 10.1007/s10957-017-1131-2. [6] H. B. Chen, Z. H. Huang and L. Q. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158. doi: 10.1007/s10589-017-9938-1. [7] H. B. Chen, L. Q. Qi and Y. S. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, (2018), 255-276. doi: 10.1007/s11464-018-0681-4. [8] H. B. Chen and Y. J. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China., 12 (2017), 1289-1302. doi: 10.1007/s11464-017-0645-0. [9] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, SIAM Series in Classics in Applied Mathematics, 2009. [10] W. Y. Ding, Z. Y. Luo and L. Q. Qi, $P$-tensors, $P_0$-tensors, and tensor complementarity problem, preprint, arXiv: 1507.06731. [11] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research and Financial Engineering, 2003. [12] G. Golub and C. Loan, Matrix Computations. Johns Hopkins series in the mathematical sciences, Johns Hopkins University Press, Baltimore, MD, 1989. [13] M. S. Gowda, Z. Y. Luo, L. Q. Qi and N. H. Xiu, $Z$-tensors and complementarity problems, preprint, arXiv: 1510.07933. [14] Z. H. Huang and L. Q. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557-576. doi: 10.1007/s10589-016-9872-7. [15] Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-Tensors, preprint, arXiv: 1509.03088. [16] M. Kojima, N. Megiddo and T. Noma, A Unified Approach to Interior-point Algorithms for Linear Complementarity Problems, in: Lecture Notes in Computer Science, vol. 538, Springer Verlag, Berlin, Germany, 1991. [17] M. Kojima, T. Noma and A. Yoshise, Global convergence in infeasible-interior-point algorithms, Math. Program., 65 (1994), 43-72. doi: 10.1007/BF01581689. [18] T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X. [19] Z. Y. Luo, L. Q. Qi and N. H. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482. doi: 10.1007/s11590-016-1013-9. [20] F. M. Ma, Y. J. Wang and H. Zhao, A potential reduction algorithm for generalized linear complementarity problem over a polyhedral cone, J. Ind. Manag. Optim., 6 (2010), 259-267. [21] H. Mansouri and M. Pirhaji, An adaptive infeasible interior-point algorithm for linear complementarity problems, J. Oper. Res. Soc., 1 (2013), 523-536. doi: 10.1007/s40305-013-0031-x. [22] M. Preiß and J. Stoer, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems, Math. Program., 99 (2004), 499-520. doi: 10.1007/s10107-003-0463-x. [23] L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. [24] L. Q. Qi, Y. J. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012. [25] L. Q. Qi, F. Wang and Y. J. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6. [26] L. Q. Qi, G. H. Yu and E. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sci., 3 (2010), 416-433. doi: 10.1137/090755138. [27] D. Savostyanov, Tensor algorithms of blind separation of electromagnetic signals, Russ. J. Numer. Anal. M., 25 (2010), 375-393. [28] E. Simantiraki and D. Shanno, An infeasible-interior-point method for linear complementarity problems, SIAM J. Optim., 7 (1997), 620-640. doi: 10.1137/S1052623495282882. [29] Y. S. Song and L. Q. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308-323. doi: 10.1007/s10957-014-0616-5. [30] Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873. doi: 10.1007/s10957-014-0616-5. [31] Y. S. Song and L. Q. Qi, Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optim. Lett., 11 (2017), 1407-1426. doi: 10.1007/s11590-016-1104-7. [32] Y. S. Song and G. H. Yu, Properties of solution set of tensor complementarity problem, J. Optim. Theory Appl., 170 (2016), 85-96. doi: 10.1007/s10957-016-0907-0. [33] K. Tanabe, Centered Newton method for mathematical programming, System Modelling Opt., 113 (1988), 197-206. [34] M. Todd and Y. Ye, A centered projective algorithm for linear programming, Math. Oper. Res., 15 (1990), 508-529. doi: 10.1287/moor.15.3.508. [35] T. Wang, R. Monteiro and J. S. Pang, An interior point potential reduction method for constrained equations, Math. Program., 74 (1996), 159-195. doi: 10.1007/BF02592210. [36] Y. J. Wang, L. Caccetta and G. L. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Lin. Alg. Appl., 22 (2015), 1059-1076. doi: 10.1002/nla.1996. [37] Y. J. Wang, L. Q. Qi and X. Z. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Lin. Alg. Appl., 16 (2009), 589-601. [38] Y. J. Wang, G. Zhou and L. Caccetta, Nonsingular $H$-tensor and its cariteria, J. Ind. Manag. Optim., 12 (2016), 1173-1186. doi: 10.3934/jimo.2016.12.1173. [39] Y. J. Wang, K. L. Zhang and H. C. Sun, Criteria for strong $H$-tensors, Front. Math. China, 11 (2016), 577-592. doi: 10.1007/s11464-016-0525-z. [40] Y. Wang, Z. H. Huang and X. L. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828. doi: 10.1080/10556788.2016.1180386. [41] S. L. Xie, D. H. Li and H. R. Xu, An iterative method for finding the least solution of the tensor complementarity problem with $Z$-Tensor, J. Optim. Theory Appl., 175 (2017), 119-136. doi: 10.1007/s10957-017-1157-5. [42] K. L. Zhang and Y. J. Wang, An $H$-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10. doi: 10.1016/j.cam.2016.03.025. [43] G. Zou, X. Chen and Z. J. Wang, Underdetermined joint blind source separation for two datasets based on tensor decomposition, IEEE Signal Proc. Lett., 23 (2016), 673-677. doi: 10.1109/LSP.2016.2546687.

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##### References:
 [1] X. L. Bai, Z. H. 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. [2] L. Castello and H. Clercx, Geometrical statistics of the vorticity vector and the strain rate tensor in rotating turbulence, J. Turbul., 14 (2013), 19-36. doi: 10.1080/14685248.2013.866241. [3] M. L. Che, L. Q. Qi and Y. M. Wei, Positive-definite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475-487. doi: 10.1007/s10957-015-0773-1. [4] H. B. Chen, Y. N. Chen, G. Y. Li and L. Q. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Lin. Alg. Appl., 25 (2018), e2125. [5] H. B. Chen, Z. H. Huang and L. Q. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optim. Theory Appl., 174 (2017), 746-761. doi: 10.1007/s10957-017-1131-2. [6] H. B. Chen, Z. H. Huang and L. Q. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158. doi: 10.1007/s10589-017-9938-1. [7] H. B. Chen, L. Q. Qi and Y. S. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, (2018), 255-276. doi: 10.1007/s11464-018-0681-4. [8] H. B. Chen and Y. J. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China., 12 (2017), 1289-1302. doi: 10.1007/s11464-017-0645-0. [9] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, SIAM Series in Classics in Applied Mathematics, 2009. [10] W. Y. Ding, Z. Y. Luo and L. Q. Qi, $P$-tensors, $P_0$-tensors, and tensor complementarity problem, preprint, arXiv: 1507.06731. [11] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research and Financial Engineering, 2003. [12] G. Golub and C. Loan, Matrix Computations. Johns Hopkins series in the mathematical sciences, Johns Hopkins University Press, Baltimore, MD, 1989. [13] M. S. Gowda, Z. Y. Luo, L. Q. Qi and N. H. Xiu, $Z$-tensors and complementarity problems, preprint, arXiv: 1510.07933. [14] Z. H. Huang and L. Q. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557-576. doi: 10.1007/s10589-016-9872-7. [15] Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-Tensors, preprint, arXiv: 1509.03088. [16] M. Kojima, N. Megiddo and T. Noma, A Unified Approach to Interior-point Algorithms for Linear Complementarity Problems, in: Lecture Notes in Computer Science, vol. 538, Springer Verlag, Berlin, Germany, 1991. [17] M. Kojima, T. Noma and A. Yoshise, Global convergence in infeasible-interior-point algorithms, Math. Program., 65 (1994), 43-72. doi: 10.1007/BF01581689. [18] T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X. [19] Z. Y. Luo, L. Q. Qi and N. H. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482. doi: 10.1007/s11590-016-1013-9. [20] F. M. Ma, Y. J. Wang and H. Zhao, A potential reduction algorithm for generalized linear complementarity problem over a polyhedral cone, J. Ind. Manag. Optim., 6 (2010), 259-267. [21] H. Mansouri and M. Pirhaji, An adaptive infeasible interior-point algorithm for linear complementarity problems, J. Oper. Res. Soc., 1 (2013), 523-536. doi: 10.1007/s40305-013-0031-x. [22] M. Preiß and J. Stoer, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems, Math. Program., 99 (2004), 499-520. doi: 10.1007/s10107-003-0463-x. [23] L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. [24] L. Q. Qi, Y. J. Wang and E. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math., 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012. [25] L. Q. Qi, F. Wang and Y. J. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6. [26] L. Q. Qi, G. H. Yu and E. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sci., 3 (2010), 416-433. doi: 10.1137/090755138. [27] D. Savostyanov, Tensor algorithms of blind separation of electromagnetic signals, Russ. J. Numer. Anal. M., 25 (2010), 375-393. [28] E. Simantiraki and D. Shanno, An infeasible-interior-point method for linear complementarity problems, SIAM J. Optim., 7 (1997), 620-640. doi: 10.1137/S1052623495282882. [29] Y. S. Song and L. Q. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308-323. doi: 10.1007/s10957-014-0616-5. [30] Y. S. Song and L. Q. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873. doi: 10.1007/s10957-014-0616-5. [31] Y. S. Song and L. Q. Qi, Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optim. Lett., 11 (2017), 1407-1426. doi: 10.1007/s11590-016-1104-7. [32] Y. S. Song and G. H. Yu, Properties of solution set of tensor complementarity problem, J. Optim. Theory Appl., 170 (2016), 85-96. doi: 10.1007/s10957-016-0907-0. [33] K. Tanabe, Centered Newton method for mathematical programming, System Modelling Opt., 113 (1988), 197-206. [34] M. Todd and Y. Ye, A centered projective algorithm for linear programming, Math. Oper. Res., 15 (1990), 508-529. doi: 10.1287/moor.15.3.508. [35] T. Wang, R. Monteiro and J. S. Pang, An interior point potential reduction method for constrained equations, Math. Program., 74 (1996), 159-195. doi: 10.1007/BF02592210. [36] Y. J. Wang, L. Caccetta and G. L. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Lin. Alg. Appl., 22 (2015), 1059-1076. doi: 10.1002/nla.1996. [37] Y. J. Wang, L. Q. Qi and X. Z. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Lin. Alg. Appl., 16 (2009), 589-601. [38] Y. J. Wang, G. Zhou and L. Caccetta, Nonsingular $H$-tensor and its cariteria, J. Ind. Manag. Optim., 12 (2016), 1173-1186. doi: 10.3934/jimo.2016.12.1173. [39] Y. J. Wang, K. L. Zhang and H. C. Sun, Criteria for strong $H$-tensors, Front. Math. China, 11 (2016), 577-592. doi: 10.1007/s11464-016-0525-z. [40] Y. Wang, Z. H. Huang and X. L. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828. doi: 10.1080/10556788.2016.1180386. [41] S. L. Xie, D. H. Li and H. R. Xu, An iterative method for finding the least solution of the tensor complementarity problem with $Z$-Tensor, J. Optim. Theory Appl., 175 (2017), 119-136. doi: 10.1007/s10957-017-1157-5. [42] K. L. Zhang and Y. J. Wang, An $H$-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10. doi: 10.1016/j.cam.2016.03.025. [43] G. Zou, X. Chen and Z. J. Wang, Underdetermined joint blind source separation for two datasets based on tensor decomposition, IEEE Signal Proc. Lett., 23 (2016), 673-677. doi: 10.1109/LSP.2016.2546687.
Numerical Results for Example 1
 $\mathit{z}^0$ $\varepsilon$ Iter Time(s) $(0.1, 0.1, 0.1, 0.1)^\top$ $10^{-5}$ 16 0.128738 $(0.1, 0.2, 0.6, 0.5)^\top$ $10^{-5}$ 20 0.181242 $(0.3, 0.5, 0.1, 0.7)^\top$ $10^{-5}$ 23 0.168388 $(0.2, 0.6, 0.4, 0.3)^\top$ $10^{-5}$ 23 0.186210 $(0.7, 0.3, 0.2, 0.5)^\top$ $10^{-5}$ 25 0.182582 $(0.2, 0.4, 0.6, 0.5)^\top$ $10^{-8}$ 35 0.194419 $(0.7, 0.5, 0.8, 0.9)^\top$ $10^{-8}$ 38 0.189600 $(1, 2, 5, 3)^\top$ $10^{-8}$ 43 0.184916 $(12, 7, 14, 35)^\top$ $10^{-8}$ 50 0.185770 $(24, 37, 56, 45)^\top$ $10^{-8}$ 55 0.221645 $(67, 52, 89, 93)^\top$ $10^{-8}$ 58 0.186433
 $\mathit{z}^0$ $\varepsilon$ Iter Time(s) $(0.1, 0.1, 0.1, 0.1)^\top$ $10^{-5}$ 16 0.128738 $(0.1, 0.2, 0.6, 0.5)^\top$ $10^{-5}$ 20 0.181242 $(0.3, 0.5, 0.1, 0.7)^\top$ $10^{-5}$ 23 0.168388 $(0.2, 0.6, 0.4, 0.3)^\top$ $10^{-5}$ 23 0.186210 $(0.7, 0.3, 0.2, 0.5)^\top$ $10^{-5}$ 25 0.182582 $(0.2, 0.4, 0.6, 0.5)^\top$ $10^{-8}$ 35 0.194419 $(0.7, 0.5, 0.8, 0.9)^\top$ $10^{-8}$ 38 0.189600 $(1, 2, 5, 3)^\top$ $10^{-8}$ 43 0.184916 $(12, 7, 14, 35)^\top$ $10^{-8}$ 50 0.185770 $(24, 37, 56, 45)^\top$ $10^{-8}$ 55 0.221645 $(67, 52, 89, 93)^\top$ $10^{-8}$ 58 0.186433
Numerical Results for Example 2
 $\mathit{z}^0$ $\varepsilon$ $\beta_0$ Iter Time(s) $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.7 91 0.311439 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.6 68 0.187435 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.5 54 0.219218 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.4 44 0.181922 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.3 37 0.176951 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.2 32 0.245196 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.1 28 0.330788 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.7 91 0.280699 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.6 67 0.232597 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.5 53 0.219453 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.4 44 0.210625 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.3 37 0.197738 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.2 32 0.169909 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.1 28 0.161405
 $\mathit{z}^0$ $\varepsilon$ $\beta_0$ Iter Time(s) $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.7 91 0.311439 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.6 68 0.187435 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.5 54 0.219218 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.4 44 0.181922 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.3 37 0.176951 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.2 32 0.245196 $(0.8147, 0.9058, 0.1270, 0.9137)^\top$ $10^{-6}$ 0.1 28 0.330788 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.7 91 0.280699 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.6 67 0.232597 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.5 53 0.219453 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.4 44 0.210625 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.3 37 0.197738 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.2 32 0.169909 $(0.9572, 0.4854, 0.8003, 0.1419)^\top$ $10^{-10}$ 0.1 28 0.161405
Numerical Results for Example 3
 $\mathit{x}^*$ Iter Time(s) $(0.0120, 0.0045, 0.0168, 0.0091, 0.0070, 0.0062)^\top$ 23 27.810409 $(0.0166, 0.0052, 0.0137, 0.0137, 0.0069, 0.0103)^\top$ 23 30.307993 $(0.0005, 0.0007, 0.0004, 0.0004, 0.0023, 0.0015)^\top$ 34 31.502331 $(0.0012, 0.0007, 0.0017, 0.0004, 0.0015, 0.0004)^\top$ 35 25.909628 $(0.0016, 0.0026, 0.0038, 0.0017, 0.0038, 0.0045)^\top$ 29 36.274344 $1.0e-003\times(0.4151, 0.1557, 0.3255, 0.0922, 0.0142, 0.3467)^\top$ 44 32.017439 $1.0e-003\times(0.0399, 0.2636, 0.3479, 0.2752, 0.4337, 0.4146)^\top$ 42 29.446150
 $\mathit{x}^*$ Iter Time(s) $(0.0120, 0.0045, 0.0168, 0.0091, 0.0070, 0.0062)^\top$ 23 27.810409 $(0.0166, 0.0052, 0.0137, 0.0137, 0.0069, 0.0103)^\top$ 23 30.307993 $(0.0005, 0.0007, 0.0004, 0.0004, 0.0023, 0.0015)^\top$ 34 31.502331 $(0.0012, 0.0007, 0.0017, 0.0004, 0.0015, 0.0004)^\top$ 35 25.909628 $(0.0016, 0.0026, 0.0038, 0.0017, 0.0038, 0.0045)^\top$ 29 36.274344 $1.0e-003\times(0.4151, 0.1557, 0.3255, 0.0922, 0.0142, 0.3467)^\top$ 44 32.017439 $1.0e-003\times(0.0399, 0.2636, 0.3479, 0.2752, 0.4337, 0.4146)^\top$ 42 29.446150
Numerical Results for Example 4
 $m$ $n$ Iter Time(s) 4 10 45 0.195758 4 20 46 1.025905 4 40 49 14.254993 4 50 50 34.858052 4 60 50 90.753663 4 80 51 465.798026 4 100 51 2702.279664 6 10 101 332.915881 6 20 123 3420.345758
 $m$ $n$ Iter Time(s) 4 10 45 0.195758 4 20 46 1.025905 4 40 49 14.254993 4 50 50 34.858052 4 60 50 90.753663 4 80 51 465.798026 4 100 51 2702.279664 6 10 101 332.915881 6 20 123 3420.345758
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