# 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, SIAM Series in Classics in Applied Mathematics, 2009.  Google Scholar [10] W. Y. Ding, Z. Y. Luo and L. Q. Qi, $P$-tensors, $P_0$-tensors, and tensor complementarity problem, preprint, arXiv: 1507.06731. Google Scholar [11] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research and Financial Engineering, 2003.  Google Scholar [12] G. Golub and C. Loan, Matrix Computations. Johns Hopkins series in the mathematical sciences, Johns Hopkins University Press, Baltimore, MD, 1989.  Google Scholar [13] M. S. Gowda, Z. Y. Luo, L. Q. Qi and N. H. Xiu, $Z$-tensors and complementarity problems, preprint, arXiv: 1510.07933. Google Scholar [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.  Google Scholar [15] Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-Tensors, preprint, arXiv: 1509.03088. Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] D. Savostyanov, Tensor algorithms of blind separation of electromagnetic signals, Russ. J. Numer. Anal. M., 25 (2010), 375-393.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] K. Tanabe, Centered Newton method for mathematical programming, System Modelling Opt., 113 (1988), 197-206.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, SIAM Series in Classics in Applied Mathematics, 2009.  Google Scholar [10] W. Y. Ding, Z. Y. Luo and L. Q. Qi, $P$-tensors, $P_0$-tensors, and tensor complementarity problem, preprint, arXiv: 1507.06731. Google Scholar [11] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research and Financial Engineering, 2003.  Google Scholar [12] G. Golub and C. Loan, Matrix Computations. Johns Hopkins series in the mathematical sciences, Johns Hopkins University Press, Baltimore, MD, 1989.  Google Scholar [13] M. S. Gowda, Z. Y. Luo, L. Q. Qi and N. H. Xiu, $Z$-tensors and complementarity problems, preprint, arXiv: 1510.07933. Google Scholar [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.  Google Scholar [15] Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-Tensors, preprint, arXiv: 1509.03088. Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] D. Savostyanov, Tensor algorithms of blind separation of electromagnetic signals, Russ. J. Numer. Anal. M., 25 (2010), 375-393.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] K. Tanabe, Centered Newton method for mathematical programming, System Modelling Opt., 113 (1988), 197-206.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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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