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Strictly semi-positive tensors and the boundedness of tensor complementarity problems

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Abstract

In this paper, we present the boundedness of solution set of tensor complementarity problem defined by a strictly semi-positive tensor. For strictly semi-positive tensor, we prove that all \(H^+(Z^+)\)-eigenvalues of each principal sub-tensor are positive. We define two new constants associated with \(H^+(Z^+)\)-eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too.

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Acknowledgements

The authors would like to thank Editor, the anonymous referees for their valuable suggestions which helped us to improve this manuscript. Our work was supported by the National Natural Science Foundation of People’s Republic of China (Grant No. 11571095, 11601134) and by the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, XinXiang, 453007, Henan, People’s Republic of China

    Yisheng Song

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi

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  1. Yisheng Song
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Correspondence to Yisheng Song.

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Song, Y., Qi, L. Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim Lett 11, 1407–1426 (2017). https://doi.org/10.1007/s11590-016-1104-7

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  • DOI: https://doi.org/10.1007/s11590-016-1104-7

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