Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 255–276 | Cite as

Column sufficient tensors and tensor complementarity problems

  • Haibin Chen
  • Liqun Qi
  • Yisheng SongEmail author
Research Article


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.


Column sufficient tensor H-eigenvalue tensor complementarity problems handicap 


65H17 15A18 90C30 


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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).


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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Management ScienceQufu Normal UniversityRizhaoChina
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongChina
  3. 3.School of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina

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