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July  2022, 18(4): 2553-2566. doi: 10.3934/jimo.2021080

Weakening convergence conditions of a potential reduction method for tensor complementarity problems

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yong Wang

Received  August 2020 Revised  January 2021 Published  July 2022 Early access  April 2021

Fund Project: The second author's work was supported by the National Natural Science Foundation of China (grant number 11871051)

Recently, under the condition that the included tensor in the tensor complementarity problem is a diagonalizable and positive definite tensor, the convergence of a potential reduction method for tensor complementarity problems is verified in [a potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2019, 15(2): 429–443]. In this paper, we improve the convergence of this method in the sense that the condition we used is strictly weaker than the one used in the above reference. Preliminary numerical results indicate the effectiveness of the potential reduction method under the new condition.

Citation: Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2553-2566. doi: 10.3934/jimo.2021080
References:
[1]

X.-L. BaiZ.-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]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer, Berlin, 2006. doi: 10.1007/978-0-387-31256-9.

[3]

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

[4] R. W. CottleJ.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992. 
[5]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 73 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.

[6]

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.

[7]

H.-B. Guan and D.-H. Li, Linearized methods for tensor complementarity problems, J. Optim. Theory Appl., 184 (2020), 972-987.  doi: 10.1007/s10957-019-01627-3.

[8]

Q. GuoM.-M. Zheng and Z.-H. Huang, Properties of $S$-tensors, Linear and Multilinear Algebra, 67 (2019), 685-696.  doi: 10.1080/03081087.2018.1430737.

[9]

L. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949-963.  doi: 10.1007/s10957-018-1422-2.

[10]

Z. H. Huang and L. 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.

[11]

Z.-H. 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.

[12]

Z.-H. 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.

[13]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer, Berlin, 1991. doi: 10.1007/3-540-54509-3.

[14]

M. KojimaT. Noma and A. Yoshise, Global convergence in infeasible-interior-point algorithms, Math. Programming, 65 (1994), 43-72.  doi: 10.1007/BF01581689.

[15]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.

[16]

D.-H. Li, C.-D. Chen and H.-B. Guan, A lower dimensional linear equation approach to the $M$-tensor complementarity problem, Calcolo, 58 (2021), Paper No. 5, 21 pp. doi: 10.1007/s10092-021-00397-7.

[17]

D. LiuW. Li and S.-W. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear and Multilinear Algebra, 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.

[18]

Z. LuoL. Qi and N. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.

[19]

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

[20]

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

[21]

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

[22]

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.

[23]

Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 3 (2017), 308-323. 

[24]

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.

[25]

D. Sun and L. Qi, On NCP-functions, Comput. Optim. Appl., 13 (1999), 201-220.  doi: 10.1023/A:1008669226453.

[26]

Y. WangZ.-H. Huang and L. Qi, Global uniqueness and solvability of tensor variational inequalities, J. Optim. Theory Appl., 177 (2018), 137-152.  doi: 10.1007/s10957-018-1233-5.

[27]

S.-L. XieD.-H. Li and H.-R. Xu, An iterative method for finding the least solution to the tensor complementarity problem, J. Optim. Theory Appl., 175 (2017), 119-136.  doi: 10.1007/s10957-017-1157-5.

[28]

K. ZhangH. 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.

show all references

References:
[1]

X.-L. BaiZ.-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]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer, Berlin, 2006. doi: 10.1007/978-0-387-31256-9.

[3]

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

[4] R. W. CottleJ.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992. 
[5]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 73 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.

[6]

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.

[7]

H.-B. Guan and D.-H. Li, Linearized methods for tensor complementarity problems, J. Optim. Theory Appl., 184 (2020), 972-987.  doi: 10.1007/s10957-019-01627-3.

[8]

Q. GuoM.-M. Zheng and Z.-H. Huang, Properties of $S$-tensors, Linear and Multilinear Algebra, 67 (2019), 685-696.  doi: 10.1080/03081087.2018.1430737.

[9]

L. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949-963.  doi: 10.1007/s10957-018-1422-2.

[10]

Z. H. Huang and L. 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.

[11]

Z.-H. 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.

[12]

Z.-H. 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.

[13]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer, Berlin, 1991. doi: 10.1007/3-540-54509-3.

[14]

M. KojimaT. Noma and A. Yoshise, Global convergence in infeasible-interior-point algorithms, Math. Programming, 65 (1994), 43-72.  doi: 10.1007/BF01581689.

[15]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.

[16]

D.-H. Li, C.-D. Chen and H.-B. Guan, A lower dimensional linear equation approach to the $M$-tensor complementarity problem, Calcolo, 58 (2021), Paper No. 5, 21 pp. doi: 10.1007/s10092-021-00397-7.

[17]

D. LiuW. Li and S.-W. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear and Multilinear Algebra, 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.

[18]

Z. LuoL. Qi and N. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.

[19]

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

[20]

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

[21]

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

[22]

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.

[23]

Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 3 (2017), 308-323. 

[24]

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.

[25]

D. Sun and L. Qi, On NCP-functions, Comput. Optim. Appl., 13 (1999), 201-220.  doi: 10.1023/A:1008669226453.

[26]

Y. WangZ.-H. Huang and L. Qi, Global uniqueness and solvability of tensor variational inequalities, J. Optim. Theory Appl., 177 (2018), 137-152.  doi: 10.1007/s10957-018-1233-5.

[27]

S.-L. XieD.-H. Li and H.-R. Xu, An iterative method for finding the least solution to the tensor complementarity problem, J. Optim. Theory Appl., 175 (2017), 119-136.  doi: 10.1007/s10957-017-1157-5.

[28]

K. ZhangH. 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.

Figure 1.  The relationships among three classes of tensors
Table 1.  Numerical Results for Example 4.1
$ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
(1.2, 0.4, 1.6280, 0.1640) 40 0.3584 (0.4642, 0.0000, 0.0000, 0.1000)
(2.4, 0.8, 13.7240, 0.6120) 46 0.3714 (0.4642, 0.0000, 0.0000, 0.1000)
(3.6, 1.2, 46.5560, 1.8280) 49 0.3776 (0.4642, 0.0000, 0.0000, 0.1000)
(4.8, 1.6,110.4920, 4.1960) 52 0.4612 (0.4642, 0.0000, 0.0000, 0.1000)
(6.0, 2.0,215.9000, 8.1000) 54 0.3880 (0.4642, 0.0000, 0.0000, 0.1000)
(7.2, 2.4,373.1480, 13.9240) 56 0.4293 (0.4642, 0.0000, 0.0000, 0.1000)
$ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
(1.2, 0.4, 1.6280, 0.1640) 40 0.3584 (0.4642, 0.0000, 0.0000, 0.1000)
(2.4, 0.8, 13.7240, 0.6120) 46 0.3714 (0.4642, 0.0000, 0.0000, 0.1000)
(3.6, 1.2, 46.5560, 1.8280) 49 0.3776 (0.4642, 0.0000, 0.0000, 0.1000)
(4.8, 1.6,110.4920, 4.1960) 52 0.4612 (0.4642, 0.0000, 0.0000, 0.1000)
(6.0, 2.0,215.9000, 8.1000) 54 0.3880 (0.4642, 0.0000, 0.0000, 0.1000)
(7.2, 2.4,373.1480, 13.9240) 56 0.4293 (0.4642, 0.0000, 0.0000, 0.1000)
Table 2.  Numerical Results for Example 4.2
$ \beta_{0} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
0.9 (0.9, 0.3, 0.6480, 0.0270) 220 1.7153 (0.0127, 0.0082, 0.0000, 0.0000)
0.8 (0.9, 0.3, 0.6480, 0.0270) 108 0.7526 (0.0125, 0.0081, 0.0000, 0.0000)
0.7 (0.9, 0.3, 0.6480, 0.0270) 70 0.5857 (0.0128, 0.0083, 0.0000, 0.0000)
0.6 (0.9, 0.3, 0.6480, 0.0270) 52 0.3878 (0.0120, 0.0078, 0.0000, 0.0000)
0.5 (0.9, 0.3, 0.6480, 0.0270) 40 0.2651 (0.0126, 0.0082, 0.0000, 0.0000)
0.4 (0.9, 0.3, 0.6480, 0.0270) 33 0.1837 (0.0116, 0.0075, 0.0000, 0.0000)
0.3 (0.9, 0.3, 0.6480, 0.0270) 27 0.2445 (0.0127, 0.0082, 0.0000, 0.0000)
0.2 (0.9, 0.3, 0.6480, 0.0270) 23 0.1956 (0.0115, 0.0075, 0.0000, 0.0000)
0.1 (0.9, 0.3, 0.6480, 0.0270) 19 0.1178 (0.0136, 0.0088, 0.0000, 0.0000)
$ \beta_{0} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
0.9 (0.9, 0.3, 0.6480, 0.0270) 220 1.7153 (0.0127, 0.0082, 0.0000, 0.0000)
0.8 (0.9, 0.3, 0.6480, 0.0270) 108 0.7526 (0.0125, 0.0081, 0.0000, 0.0000)
0.7 (0.9, 0.3, 0.6480, 0.0270) 70 0.5857 (0.0128, 0.0083, 0.0000, 0.0000)
0.6 (0.9, 0.3, 0.6480, 0.0270) 52 0.3878 (0.0120, 0.0078, 0.0000, 0.0000)
0.5 (0.9, 0.3, 0.6480, 0.0270) 40 0.2651 (0.0126, 0.0082, 0.0000, 0.0000)
0.4 (0.9, 0.3, 0.6480, 0.0270) 33 0.1837 (0.0116, 0.0075, 0.0000, 0.0000)
0.3 (0.9, 0.3, 0.6480, 0.0270) 27 0.2445 (0.0127, 0.0082, 0.0000, 0.0000)
0.2 (0.9, 0.3, 0.6480, 0.0270) 23 0.1956 (0.0115, 0.0075, 0.0000, 0.0000)
0.1 (0.9, 0.3, 0.6480, 0.0270) 19 0.1178 (0.0136, 0.0088, 0.0000, 0.0000)
Table 3.  Numerical Results for Example 4.3
$ {q}^{\top} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
$ (-1, 1) $ (2, 1, 28, 2) 47 0.3207 (1, 0, 0, 1)
$ (-1, 1) $ (4.2, 2.1, 1183.3893, 41.8410) 67 0.4496 (1, 0, 0, 1)
$ (-6, -2) $ (2.8, 1.4,149.9690, 3.3782) 52 0.3200 (1.6438, 1.1487, 0, 0)
$ (-6, -2) $ (8, 4, 29690, 1022) 129 1.3179 (1.6438, 1.1487, 0, 0)
$ (36, -19) $ (4, 2,964, 13) 66 0.4528 (1.8384, 1.8020, 0, 0)
$ (36, -19) $ (24, 12, 7216164, 248813) 760 7.4286 (1.8384, 1.8020, 0, 0)
$ {q}^{\top} $ $ (z^{0})^{\top} $ $ \mathrm{Iter} $ $ \mathrm{Time(s)} $ $ (z^{*})^{\top} $
$ (-1, 1) $ (2, 1, 28, 2) 47 0.3207 (1, 0, 0, 1)
$ (-1, 1) $ (4.2, 2.1, 1183.3893, 41.8410) 67 0.4496 (1, 0, 0, 1)
$ (-6, -2) $ (2.8, 1.4,149.9690, 3.3782) 52 0.3200 (1.6438, 1.1487, 0, 0)
$ (-6, -2) $ (8, 4, 29690, 1022) 129 1.3179 (1.6438, 1.1487, 0, 0)
$ (36, -19) $ (4, 2,964, 13) 66 0.4528 (1.8384, 1.8020, 0, 0)
$ (36, -19) $ (24, 12, 7216164, 248813) 760 7.4286 (1.8384, 1.8020, 0, 0)
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