Column sufficient tensors and tensor complementarity problems
- 43 Downloads
Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.
KeywordsColumn sufficient tensor H-eigenvalue tensor complementarity problems handicap
MSC65H17 15A18 90C30
Unable to display preview. Download preview PDF.
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11601261, 11571095, 11601134), the Hong Kong Research Grant Council (Grant No.PolyU 502111, 501212, 501913, 15302114), the Natural Science Foundation of Shandong Province (No. ZR2016AQ12), and the China Postdoctoral Science Foundation (Grant No. 2017M622163).
- 6.Chen H, Huang Z, Qi L. Copositive tensor detection and its applications in physics and hypergraphs. Comput Optim Appl, 2017, https://doi.org/10.1007/s10589-017-9938-1Google Scholar
- 12.Cottle R W, Guu S M. Are P *-matrices just sufficient? Presented at the 36th Joint National Meeting of Operations Research Society of America and the Institute of Management Science, Phoenix, AZ, 1 Nov, 1993Google Scholar
- 17.Gowda M S, Luo Z, Qi L, Xiu N. Z-tensors and complementarity problems. 2015, arXiv: 1510.07933Google Scholar
- 18.Guu S M, Cottle R W. On a subclass of P 0: Linear Algebra Appl, 1995, 223: 325–335Google Scholar
- 19.Han J Y, Xiu N H, Qi H D. Nonlinear Complementary Theory and Algorithm. Shanghai: Shanghai Science and Technology Press, 2006Google Scholar
- 26.Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP05), Vol 1. 2005, 129–132Google Scholar
- 42.Song Y, Qi L. Properties of tensor complementarity problem and some classes of structured tensors. Ann of Appl Math, 2017, 33(3): 308–323Google Scholar