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April  2019, 15(2): 933-946. doi: 10.3934/jimo.2018078

Global error bounds for the tensor complementarity problem with a P-tensor

Mengmeng Zheng , Ying Zhang and  Zheng-Hai Huang ,

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Zheng-Hai Huang

Received  September 2017 Revised  January 2018 Published  April 2019 Early access  June 2018

Fund Project: The third author is supported by the National Natural Science Foundation of China (Grant No. 11431002).

As a natural extension of the linear complementarity problem, the tensor complementarity problem has been studied recently; and many theoretical results have been obtained. In this paper, we investigate the global error bound for the tensor complementarity problem with a P-tensor. We give two global error bounds for this class of complementarity problems with the help of two positively homogeneous operators defined by a P-tensor. When the order of the involved tensor reduces to 2, the results obtained in this paper coincide exactly with the one for the linear complementarity problem.

Keywords: Tensor complementarity problem, global error bound, P-tensor, positively homogeneous operator, linear complementarity problem.
Mathematics Subject Classification: Primary: 90C33, 65K10; Secondary: 15A69.
Citation: Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial & Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078
References:
[1]

X. L. Bai, Z. H. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, Journal of Optimization Theory and Applications, 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrix in the Mathematical Sciences, Society for Industrial and Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

M. L. Che, L. Qi and Y. M. Wei, Positive definite tensors to nonlinear complementarity problems, Journal of Optimization Theory and Applications, 168 (2016), 475-487.  doi: 10.1007/s10957-015-0773-1.  Google Scholar

[4]

T. T. Chen, W. Li, X. P. Wu and S. Vong, Error bounds for linear complementarity problems of MB-matrices, Numerical Algorithms, 70 (2015), 341-356.  doi: 10.1007/s11075-014-9950-9.  Google Scholar

[5]

X. Chen and S. Xiang, Computation of error bounds for P-matrix linear complementarity problems, Mathematical Programming, 106 (2006), 513-525.  doi: 10.1007/s10107-005-0645-9.  Google Scholar

[6]

X. Chen and S. Xiang, Perturbation bounds of P-matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 1250-1265.  doi: 10.1137/060653019.  Google Scholar

[7]

R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992.  Google Scholar

[8]

P. F. Dai, Error bounds for linear complementarity problems of DB-matrices, Linear Algebra and its Applications, 434 (2011), 830-840.  doi: 10.1016/j.laa.2010.09.049.  Google Scholar

[9]

P. F. Dai, Y. T. Li and C. J. Lu, Error bounds for linear complementarity problems for SB-matrices, Numerical Algorithms, 61 (2012), 121-139.  doi: 10.1007/s11075-012-9533-6.  Google Scholar

[10]

P. F. Dai, Y. T. Li and C. J. Lu, New error bounds for linear complementarity problem with an SB-matrices, Numerical Algorithms, 64 (2013), 741-757.  doi: 10.1007/s11075-012-9691-6.  Google Scholar

[11]

L. Gao and C. Li, An improved error bound for linear complementarity problems for B-matrices, Journal of Inequalities and Applications, (2017), Paper No. 144, 10 pp. doi: 10.1186/s13660-017-1414-z.  Google Scholar

[12]

M. García-Esnaola and J. M. Peña, Error bounds for linear complementarity problems for $B$-matrices, Applied Mathematics Letters, 22 (2009), 1071-1075.  doi: 10.1016/j.aml.2008.09.001.  Google Scholar

[13]

M. S. Gowda, Z. Y. Luo, L. Qi and N. H. Xiu, Z tensors and complementarity problems, arXiv: 1510.07933v2 (2016). Google Scholar

[14]

Q. Guo, M. M. Zheng and Z. H. Huang, Properties of S-tensor Linear and Multilinear Algebra, (2018). doi: 10.1080/03081087.2018.1430737.  Google Scholar

[15]

Z. H. Huang and L. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Computational Optimization and Applications, 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.  Google Scholar

[16]

Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-tensors, to appear in Pacific Journal of Optimization, arXiv: 1509.03088 (2015). Google Scholar

[17]

C. Li, M. Gan and S. Yang, A new error bound for linear complementarity problems for B-matrices, Electronic Journal of Linear Algebra, 31 (2016), 476-484.  doi: 10.13001/1081-3810.3250.  Google Scholar

[18]

W. Li and H. Zheng, Some new error bounds for linear complementarity problems of H-matrices, Numerical Algorithms, 67 (2014), 257-269.  doi: 10.1007/s11075-013-9786-8.  Google Scholar

[19]

Z. Q. Luo, O. L. Mangasarian, J. Ren and M. V. Solodov, New error bounds for the linear complementarity problem, Mathematics of Operations Research, 19 (1994), 880-892.  doi: 10.1287/moor.19.4.880.  Google Scholar

[20]

Z. Y. Luo, L. Qi and N. H. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optimization Letters, 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.  Google Scholar

[21]

R. Mathias and J.-S. Pang, Error bounds for the linear complementarity problem with a P-matrix, Linear Algebra and its Applications, 132 (1990), 123-136.  doi: 10.1016/0024-3795(90)90058-K.  Google Scholar

[22]

K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988.  Google Scholar

[23]

L. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial & Applied Mathematics, U. S. A, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[24]

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

[25]

Y. S. Song and L. Qi, Tensor complementarity problem and semi-positive tensors, Journal of Optimization Theory and Applications, 169 (2016), 1069-1078.  doi: 10.1007/s10957-015-0800-2.  Google Scholar

[26]

Y. S. Song and L. Qi, Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optimization Letters, 11 (2017), 1407-1426.  doi: 10.1007/s11590-016-1104-7.  Google Scholar

[27]

Y. S. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Annals of Applied Mathematics, 33 (2017), 308-323.   Google Scholar

[28]

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

[29]

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

[30]

W. Yu, C. Ling, H. J. He and L. Qi, On the properties of tensor complementarity problems, arXiv: 1608.01735v1 (2016). Google Scholar

[31]

P. Z. Yuan and L. H. You, Some remarks on P, P0, B and B0 tensors, Linear Algebra and its Applications, 459 (2014), 511-521.  doi: 10.1016/j.laa.2014.07.043.  Google Scholar

show all references

References:
[1]

X. L. Bai, Z. H. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, Journal of Optimization Theory and Applications, 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrix in the Mathematical Sciences, Society for Industrial and Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

M. L. Che, L. Qi and Y. M. Wei, Positive definite tensors to nonlinear complementarity problems, Journal of Optimization Theory and Applications, 168 (2016), 475-487.  doi: 10.1007/s10957-015-0773-1.  Google Scholar

[4]

T. T. Chen, W. Li, X. P. Wu and S. Vong, Error bounds for linear complementarity problems of MB-matrices, Numerical Algorithms, 70 (2015), 341-356.  doi: 10.1007/s11075-014-9950-9.  Google Scholar

[5]

X. Chen and S. Xiang, Computation of error bounds for P-matrix linear complementarity problems, Mathematical Programming, 106 (2006), 513-525.  doi: 10.1007/s10107-005-0645-9.  Google Scholar

[6]

X. Chen and S. Xiang, Perturbation bounds of P-matrix linear complementarity problems, SIAM Journal on Optimization, 18 (2007), 1250-1265.  doi: 10.1137/060653019.  Google Scholar

[7]

R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992.  Google Scholar

[8]

P. F. Dai, Error bounds for linear complementarity problems of DB-matrices, Linear Algebra and its Applications, 434 (2011), 830-840.  doi: 10.1016/j.laa.2010.09.049.  Google Scholar

[9]

P. F. Dai, Y. T. Li and C. J. Lu, Error bounds for linear complementarity problems for SB-matrices, Numerical Algorithms, 61 (2012), 121-139.  doi: 10.1007/s11075-012-9533-6.  Google Scholar

[10]

P. F. Dai, Y. T. Li and C. J. Lu, New error bounds for linear complementarity problem with an SB-matrices, Numerical Algorithms, 64 (2013), 741-757.  doi: 10.1007/s11075-012-9691-6.  Google Scholar

[11]

L. Gao and C. Li, An improved error bound for linear complementarity problems for B-matrices, Journal of Inequalities and Applications, (2017), Paper No. 144, 10 pp. doi: 10.1186/s13660-017-1414-z.  Google Scholar

[12]

M. García-Esnaola and J. M. Peña, Error bounds for linear complementarity problems for $B$-matrices, Applied Mathematics Letters, 22 (2009), 1071-1075.  doi: 10.1016/j.aml.2008.09.001.  Google Scholar

[13]

M. S. Gowda, Z. Y. Luo, L. Qi and N. H. Xiu, Z tensors and complementarity problems, arXiv: 1510.07933v2 (2016). Google Scholar

[14]

Q. Guo, M. M. Zheng and Z. H. Huang, Properties of S-tensor Linear and Multilinear Algebra, (2018). doi: 10.1080/03081087.2018.1430737.  Google Scholar

[15]

Z. H. Huang and L. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Computational Optimization and Applications, 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.  Google Scholar

[16]

Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-tensors, to appear in Pacific Journal of Optimization, arXiv: 1509.03088 (2015). Google Scholar

[17]

C. Li, M. Gan and S. Yang, A new error bound for linear complementarity problems for B-matrices, Electronic Journal of Linear Algebra, 31 (2016), 476-484.  doi: 10.13001/1081-3810.3250.  Google Scholar

[18]

W. Li and H. Zheng, Some new error bounds for linear complementarity problems of H-matrices, Numerical Algorithms, 67 (2014), 257-269.  doi: 10.1007/s11075-013-9786-8.  Google Scholar

[19]

Z. Q. Luo, O. L. Mangasarian, J. Ren and M. V. Solodov, New error bounds for the linear complementarity problem, Mathematics of Operations Research, 19 (1994), 880-892.  doi: 10.1287/moor.19.4.880.  Google Scholar

[20]

Z. Y. Luo, L. Qi and N. H. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optimization Letters, 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.  Google Scholar

[21]

R. Mathias and J.-S. Pang, Error bounds for the linear complementarity problem with a P-matrix, Linear Algebra and its Applications, 132 (1990), 123-136.  doi: 10.1016/0024-3795(90)90058-K.  Google Scholar

[22]

K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988.  Google Scholar

[23]

L. Qi and Z. Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial & Applied Mathematics, U. S. A, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[24]

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

[25]

Y. S. Song and L. Qi, Tensor complementarity problem and semi-positive tensors, Journal of Optimization Theory and Applications, 169 (2016), 1069-1078.  doi: 10.1007/s10957-015-0800-2.  Google Scholar

[26]

Y. S. Song and L. Qi, Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optimization Letters, 11 (2017), 1407-1426.  doi: 10.1007/s11590-016-1104-7.  Google Scholar

[27]

Y. S. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Annals of Applied Mathematics, 33 (2017), 308-323.   Google Scholar

[28]

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

[29]

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

[30]

W. Yu, C. Ling, H. J. He and L. Qi, On the properties of tensor complementarity problems, arXiv: 1608.01735v1 (2016). Google Scholar

[31]

P. Z. Yuan and L. H. You, Some remarks on P, P0, B and B0 tensors, Linear Algebra and its Applications, 459 (2014), 511-521.  doi: 10.1016/j.laa.2014.07.043.  Google Scholar

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