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Properties of Structured Tensors and Complementarity Problems

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Abstract

In this paper, we present some new results on a class of tensors, which are defined by the solvability of the corresponding tensor complementarity problem. For such structured tensors, we give a sufficient condition to guarantee the nonzero solution of the corresponding tensor complementarity problem with a vector containing at least two nonzero components and discuss their relationships with some other structured tensors. Furthermore, with respect to the tensor complementarity problem with a nonnegative such structured tensor, we obtain the upper and lower bounds of its solution set, and by the way, we show that the eigenvalues of such a tensor are closely related to this solution set.

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Acknowledgements

The authors would like to thank the anonymous referees/editors 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. 11671217).

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Correspondence to Qingzhi Yang.

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Communicated by Liqun Qi.

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Mei, W., Yang, Q. Properties of Structured Tensors and Complementarity Problems. J Optim Theory Appl (2020). https://doi.org/10.1007/s10957-020-01631-y

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Keywords

  • Structured tensor
  • Tensor complementarity problems
  • Strictly semi-positive tensor
  • Norm
  • Upper and lower bounds

Mathematics Subject Classification

  • 47H15
  • 47H12
  • 34B10
  • 47A52
  • 47J10
  • 47H09
  • 15A48
  • 47H07