Journal of the Operations Research Society of China >

2025 , Vol. 13 >Issue 4: 946 - 965

DOI: https://doi.org/10.1007/s40305-023-00531-y

Some Properties of the Solution of the Extended Vertical Tensor Complementarity Problem

Expand
  • 1 School of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, China;
    2 Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming 650500, Yunnan, China

Received date: 2023-04-14

  Revised date: 2023-11-14

  Online published: 2025-12-19

Supported by

This research was supported by the National Natural Science Foundations of China (No. 11961082)

Fold

Abstract

In this paper, we introduce the extended vertical tensor complementarity problem and investigate the existence and uniqueness of its solution. We introduce several sets of special tensors and demonstrate their properties. We leverage the structure of tensors to obtain some properties of the solution of the extended vertical tensor complementarity problem. Furthermore, we use degree theory and the equivalent form of the minimum function to establish sufficient conditions for the existence and uniqueness of the solution of the extended vertical tensor complementarity problem, respectively.

Key words： The extended vertical tensor complementarity problem; Special tensor sets; The degree theory; The minimum function

Cite this article

Li-Ming Li, Shi-Liang Wu, Cui-Xia Li . Some Properties of the Solution of the Extended Vertical Tensor Complementarity Problem[J]. Journal of the Operations Research Society of China, 2025 , 13(4) : 946 -965 . DOI: 10.1007/s40305-023-00531-y

References

[1] Song, Y.-S., Qi, L.-Q.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33, 308–323 (2017)
[2] Qi, L.-Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
[3] Che, M.-L., Qi, L.-Q., Wei, Y.-M.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016)
[4] Gowda, M.S., Luo, Z.-Y., Qi, L.-Q., Xiu, N.-H.: Z-tensors and complementarity problems. arXiv:1510.07933 (2015)
[5] Huang, Z.-H., Qi, L.-Q.: Formulating an n-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017)
[6] Huang, Z.-H., Suo, Y.-Y., Wang, J.: On Q-tensors. arXiv:1509.03088 (2015)
[7] Luo, Z.-Y., Qi, L.-Q., Xiu, N.-H.: The sparsest solutions to Z-tensor complementarity problems. Optim. Lett. 11, 471–482 (2017)
[8] Song, Y.-S., Qi, L.-Q.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)
[9] Bai, X.-L., Huang, Z.-H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)
[10] Dai, P.-F., Wu, S.-L.: The GUS-property and modulus-based methods for tensor complementarity problems. J. Optim. Theory Appl. 195, 976–1006 (2022)
[11] Du, S.-Q., Zhang, L.-P.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019)
[12] Du, S.-Q., Zhang, L.-P., Chen, C.-Y., Qi, L.-Q.: Tensor absolute value equations. Sci. China Math. 61, 1695–1710 (2018)
[13] Han, L.-X.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949–963 (2019)
[14] Song, Y.-S., Qi, L.-Q.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016)
[15] Liu, D.-D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear A. 66, 1726–1749 (2018)
[16] Wang, X.-Z., Che, M.-L., Wei, Y.-M.: Randomized Kaczmarz methods for tensor complementarity problems. Comput. Optim. Appl. 82, 595–615 (2022)
[17] Che, M.-L., Qi, L.-Q., Wei, Y.-M.: The generalized order tensor complementarity problems. Numer. Math. Theory Methods Appl. 13, 131–149 (2020)
[18] Gowda, M.S., Sznajder, R.: The generalized order linear complementarity problem. SIAM J. Matrix Anal. Appl. 15, 779–795 (1994)
[19] Sharma, S., Palpandi, K.: Some existence results for the generalized tensor absolute value equation. Filomat 37, 4185–4194 (2023)
[20] Shao, J.-Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350–2366 (2013)
[21] Lloyd, N.G.: Degree Theory. Cambridge University Press, London (1978)
[22] Ding, W.-Y., Qi, L.-Q., Wei, Y.-M.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)
[23] Wu, S.-L., Li, W., Wang, H.-H.: The perturbation bound of the extended vertical linear complementarity problem. J. Oper. Res. Soc. China (2023). https://doi.org/10.1007/s40305-023-00456-6
[24] Ding, W.-Y., Wei, Y.-M.: Solving multi-linear systems with M-tensors. J. Sci. Comput. 68, 689–715 (2016)
Options
Outlines

/

〈 〉

Copyright © Operations Research Society of China, The Periodicals Agency of Shanghai University , and Springer-Verlag Berlin Heidelberg 2016