Abstract
Recently, the tensor complementarity problem has been investigated in the literature. In this paper, we extend a class of structured matrices to higher-order tensors; the corresponding tensor complementarity problem has a unique solution for any nonzero nonnegative vector. We discuss their relationships with semi-positive tensors and strictly semi-positive tensors. We also study the property of such a structured tensor. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. We also give two equivalent formulations of such a structured tensor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)
Pang, J.S.: On Q-matrices. Math. Program. 17, 243–247 (1979)
Gowda, M.S.: On Q-matrices. Math. Program. 49, 139–141 (1990)
Jeter, M.W., Pye, W.C.: An example of a nonregular semimonotone Q-matrix. Math. Program. 44, 351–356 (1989)
Murthy, G.S.R., Parthasarathy, T., Ravindran, G.: A copositive Q-matrix which is not \(R_0\). Math Program. 61, 131–135 (1993)
Eaves, B.C.: The linear complementarity problem. Manag. Sci. 17, 621–634 (1971)
Danao, R.A.: On a class of semimonotone \(Q_0\)-matrices in the linear complementarity problem. Oper. Res. Lett. 13, 121–125 (1993)
Danao, R.A.: A note on \(E^{\prime }\)-matrices. Linear Algebra Appl. 259, 299–305 (1997)
Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33(3), 308–323 (2017)
Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)
Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169(3), 1069–1078 (2016)
Cottle, R.W.: Completely Q-matrices. Math. Program. 19, 347–351 (1980)
Chen, H., Huang, Z., Qi, L.: Copositivity detection of tensors: theory and algorithm. J. Optim. Theory Appl. 174, 746–761 (2017)
Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
Song, Y., Qi, L.: Necessary and sufficient conditions for copositive tensors. Linear Multilinear Algebra 63(1), 120–131 (2015)
Acknowledgements
The authors would like to thank the editors and anonymous referees for their valuable suggestions which helped us to improve this manuscript. This work was supported by the National Natural Science Foundation of China (Grant No. 11371276).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Liqun Qi.
Rights and permissions
About this article
Cite this article
Zheng, Yn., Wu, W. On a Class of Semi-Positive Tensors in Tensor Complementarity Problem. J Optim Theory Appl 177, 127–136 (2018). https://doi.org/10.1007/s10957-018-1262-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-018-1262-0