An error bound estimation for the positive semi-definite tensor complementarity problem and its applications

Yuanshou Zhang; Hongchun Sun; Zhiwen Jie; Sabir Amina; Yuanshou Zhang; Hongchun Sun; Zhiwen Jie; Sabir Amina

doi:10.3934/math.2025929

AIMS Mathematics

2025, Volume 10, Issue 9: 20805-20824. doi: 10.3934/math.2025929

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Research article

An error bound estimation for the positive semi-definite tensor complementarity problem and its applications

1.
School of Continuing Education, Weifang University, Weifang 261000, China
2.
School of Mathematics and Statistics, Linyi University, Linyi 276005, China
3.
School of Mathematics and Statistics, Kashi University, Kashi 844006, China

Received: 25 May 2025 Revised: 10 August 2025 Accepted: 25 August 2025 Published: 09 September 2025
MSC : 90C30, 90C33

For the positive semi-definite tensor complementarity problem (TCP), based on the natural residual function, we first established an error bound estimation for the positive semi-definite TCP without the fractional term of the residual function. Compared with the existing results, the requirements imposed on the TCP such as being an $ m $-uniform $ P $-function and being $ m $-monotone were removed. As an application of the error bound obtained, we showed the global $ R $-linear convergence rate of the proposed self-adaptive projection algorithm for solving the TCP via an equivalent transformation of this problem. Meanwhile, we also obtained an $ \epsilon $-optimal solution in a finite number of iterations. Finally, numerical results were reported to demonstrate the efficiency of the proposed method.
- positive semi-definite TCP,
- error bound,
- self-adaptive projection algorithm,
- global $ R $-linear convergence rate
Citation: Yuanshou Zhang, Hongchun Sun, Zhiwen Jie, Sabir Amina. An error bound estimation for the positive semi-definite tensor complementarity problem and its applications[J]. AIMS Mathematics, 2025, 10(9): 20805-20824. doi: 10.3934/math.2025929

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Abstract

For the positive semi-definite tensor complementarity problem (TCP), based on the natural residual function, we first established an error bound estimation for the positive semi-definite TCP without the fractional term of the residual function. Compared with the existing results, the requirements imposed on the TCP such as being an $ m $-uniform $ P $-function and being $ m $-monotone were removed. As an application of the error bound obtained, we showed the global $ R $-linear convergence rate of the proposed self-adaptive projection algorithm for solving the TCP via an equivalent transformation of this problem. Meanwhile, we also obtained an $ \epsilon $-optimal solution in a finite number of iterations. Finally, numerical results were reported to demonstrate the efficiency of the proposed method.

References

[1]	Y. S. Song, L. Q. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854–873. https://doi.org/10.1007/s10957-014-0616-5 doi: 10.1007/s10957-014-0616-5
[2]	Y. S. Song, L. Q. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308–323.
[3]	M. L. Che, L. Q. Qi, Y. M. Wei, Positive-definite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475–487. https://doi.org/10.1007/s10957-015-0773-1 doi: 10.1007/s10957-015-0773-1
[4]	Z. H. Huang, L. Q. Qi, Tensor complementarity problems–part III: Applications, J. Optim. Theory Appl., 183 (2019), 771–791. https://doi.org/10.1007/s10957-019-01573-0 doi: 10.1007/s10957-019-01573-0
[5]	L. Q. Qi, H. B. Chen, Y. N. Chen, Tensor eigenvalues and their applications, Singapore: Springer, 2018. https://doi.org/10.1007/978-981-10-8058-6
[6]	Z. H. Huang, L. Q. Qi, Formulating an $n$-person noncoorperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557–576. https://doi.org/10.1007/s10589-016-9872-7 doi: 10.1007/s10589-016-9872-7
[7]	Z. Y. Luo, L. Q. Qi, N. H. Xiu, The sparsest solutions to Z-tensor complementarity problems, Optim. Lett., 11 (2017), 471–482. https://doi.org/10.1007/s11590-016-1013-9 doi: 10.1007/s11590-016-1013-9
[8]	Z. H. Huang, L. Q. Qi, Tensor complementarity problems–part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1–23. https://doi.org/10.1007/s10957-019-01566-z doi: 10.1007/s10957-019-01566-z
[9]	M. M. Zheng, Y. Zhang, Z. H. Huang, Global error bounds for the tensor complementarity problem with a P-tensor, J. Ind. Manag. Optim., 15 (2019), 933–946. https://doi.org/10.3934/jimo.2018078 doi: 10.3934/jimo.2018078
[10]	L. Y. Ling, C. Ling, H. J. He, Generalized tensor complementarity problems over a polyhedral cone, Asia Pac. J. Oper. Res., 37 (2020), 2040006. https://doi.org/10.1142/S0217595920400060 doi: 10.1142/S0217595920400060
[11]	L. Y. Ling, H. J. He, C. Ling, On error bounds of polynomial complementarity problems with structured tensors, Optimization, 67 (2018), 341–358. https://doi.org/10.1080/02331934.2017.1391254 doi: 10.1080/02331934.2017.1391254
[12]	S. L. Hu, J. Wang, Z. H. Huang, Error bounds for the solution sets of quadratic complementarity problems, J. Optim. Theory Appl., 179 (2018), 983–1000. https://doi.org/10.1007/s10957-018-1356-8 doi: 10.1007/s10957-018-1356-8
[13]	L. Q. Qi, Z. H. Huang, Tensor complementarity problems–part II: Solution methods, J. Optim. Theory Appl., 183 (2019), 365–385. https://doi.org/10.1007/s10957-019-01568-x doi: 10.1007/s10957-019-01568-x
[14]	L. X. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949–963. https://doi.org/10.1007/s10957-018-1422-2 doi: 10.1007/s10957-018-1422-2
[15]	K. L. Zhang, H. B. Chen, P. F. Zhao, A potential reduction method for tensor complementarity problems, J. Ind. Manag. Optim., 15 (2019), 429–443. https://doi.org/10.3934/jimo.2018049 doi: 10.3934/jimo.2018049
[16]	D. D. Liu, W. Li, S. W. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear Multilinear Algebra, 66 (2018), 1726–1749. https://doi.org/10.1080/03081087.2017.1369929 doi: 10.1080/03081087.2017.1369929
[17]	F. Y. Han, Y. M. Wei, P. P. Xie, Regularized and structured tensor total least squares methods with applications, J. Optim. Theory Appl., 202 (2024), 1101–1136. https://doi.org/10.1007/s10957-024-02507-1 doi: 10.1007/s10957-024-02507-1
[18]	X. Z. Wang, M. L. Che, L. Q. Qi, Y. M. Wei, Modified gradient dynamic approach to the tensor complementarity problem, Optim. Method Softw., 35 (2020), 394–415. https://doi.org/10.1080/10556788.2019.1578766 doi: 10.1080/10556788.2019.1578766
[19]	J. J. Sun, S. Q. Du, An effective smoothing Newton projection algorithm for finding sparse solutions to NP-hard tensor complementarity problems, J. Comput. Appl. Math., 451 (2024), 116074. https://doi.org/10.1016/j.cam.2024.116074 doi: 10.1016/j.cam.2024.116074
[20]	P. F. Dai, A fixed point iterative method for tensor complementarity problems, J. Sci. Comput., 84 (2020), 49. https://doi.org/10.1007/s10915-020-01299-6 doi: 10.1007/s10915-020-01299-6
[21]	G. Li, J. C. Li, Improved fixed point iterative methods for tensor complementarity problem, J. Optim. Theory Appl., 199 (2023), 787–804. https://doi.org/10.1007/s10957-023-02304-2 doi: 10.1007/s10957-023-02304-2
[22]	L. Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302–1324. https://doi.org/10.1016/j.jsc.2005.05.007 doi: 10.1016/j.jsc.2005.05.007
[23]	G. H. Golub, C. F. Van Loan, Matrix computations, 4 Eds., Johns Hopkins University Press, 2013.
[24]	W. Y. Ding, Y. M. Wei, Generalized tensor eigenvalue problems, SIAM J. Matrix Anal. Appl., 36 (2015), 1073–1099. https://doi.org/10.1137/140975656 doi: 10.1137/140975656
[25]	E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory: Part I. Projections on convex sets: Part II. Spectral theory, In: Contributions to nonlinear functional analysis, New York: Academic Press, 1971,237–424. https://doi.org/10.1016/B978-0-12-775850-3.50013-3
[26]	M. A. Noor, General variational inequalities, Appl. Math. Lett., 1 (1988), 119–122. https://doi.org/ 10.1016/0893-9659(88)90054-7 doi: 10.1016/0893-9659(88)90054-7
[27]	J. J. Duistermaat, J. A. C. Kolk, Multidimensional real analysis I: Differentiation, New York: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511616716
[28]	Y. S. Song, L. Q. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581–1595. https://doi.org/10.1137/130909135 doi: 10.1137/130909135
[29]	Y. S. Song, L. Q. Qi, Strictly semi-positive tensors and the boundedness of tensor complementarity problems, Optim. Lett., 11 (2017), 1407–1426. https://doi.org/10.48550/arXiv.1509.01327 doi: 10.48550/arXiv.1509.01327
[30]	T. T. Shang, G. J. Tang, Structured tensor tuples to polynomial complementarity problems, J. Global Optim., 86 (2023), 867–883. https://doi.org/10.1007/s10898-023-01302-y doi: 10.1007/s10898-023-01302-y
[31]	Y. Xu, G. Y. Ni, M. S. Zhang, Bounds of the solution set to the polynomial complementarity problem, J. Optim. Theory Appl., 203 (2024), 146–164. https://doi.org/10.1007/s10957-024-02484-5 doi: 10.1007/s10957-024-02484-5
[32]	T. T. Shang, W. S. Jia, Q tensor tuples to polynomial complementarity problems, Oper. Res. Lett., 61 (2025), 107297. https://doi.org/10.1016/j.orl.2025.107297 doi: 10.1016/j.orl.2025.107297
[33]	F. Facchinei, J. S. Pang, Finite-dimensional variational inequality and complementarity problems, 1 Eds., New York: Springer, 2003. https://doi.org/10.1007/b97544

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