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Global error bounds for the tensor complementarity problem with a P-tensor
School of Mathematics, Tianjin University, Tianjin 300350, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. W. Cottle, J.-S. Pang and R. E. Stone,
The Linear Complementarity Problem, Academic Press, Boston, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
M. S. Gowda, Z. Y. Luo, L. Qi and N. H. Xiu,
Z tensors and complementarity problems, arXiv: 1510.07933v2 (2016). |
[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. |
[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. |
[16] |
Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-tensors, to appear in Pacific Journal of Optimization, arXiv: 1509.03088 (2015). |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
K. G. Murty,
Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[30] |
W. Yu, C. Ling, H. J. He and L. Qi, On the properties of tensor complementarity problems, arXiv: 1608.01735v1 (2016). |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. W. Cottle, J.-S. Pang and R. E. Stone,
The Linear Complementarity Problem, Academic Press, Boston, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
M. S. Gowda, Z. Y. Luo, L. Qi and N. H. Xiu,
Z tensors and complementarity problems, arXiv: 1510.07933v2 (2016). |
[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. |
[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. |
[16] |
Z. H. Huang, Y. Y. Suo and J. Wang, On $Q$-tensors, to appear in Pacific Journal of Optimization, arXiv: 1509.03088 (2015). |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
K. G. Murty,
Linear Complementarity, Linear and Nonlinear Programming, Heldermann, Berlin, 1988. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[30] |
W. Yu, C. Ling, H. J. He and L. Qi, On the properties of tensor complementarity problems, arXiv: 1608.01735v1 (2016). |
[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. |
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