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Tensor Complementarity Problems—Part I: Basic Theory

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Abstract

Tensors (hypermatrices) are multidimensional analogs of matrices. The tensor complementarity problem is a class of nonlinear complementarity problems with the involved function being defined by a tensor, which is also a direct and natural extension of the linear complementarity problem. In the last few years, the tensor complementarity problem has attracted a lot of attention, and has been studied extensively, from theory to solution methods and applications. This work, with its three parts, aims at contributing to review the state-of-the-art of studies for the tensor complementarity problem and related models. In this part, we describe the theoretical developments for the tensor complementarity problem and related models, including the nonemptiness and compactness of the solution set, global uniqueness and solvability, error bound theory, stability and continuity analysis, and so on. The developments of solution methods and applications for the tensor complementarity problem are given in the second part and the third part, respectively. Some further issues are proposed in all the parts.

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Acknowledgements

We are very grateful to professors Chen Ling, Yisheng Song, Shenglong Hu and Ziyan Luo for reading the first draft of this paper and putting forward valuable suggestions for revision. The first author’s work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002 and 11871051), and the second author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717).

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  1. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Zheng-Hai Huang

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

    Liqun Qi

  3. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Liqun Qi

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  1. Zheng-Hai Huang
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Huang, ZH., Qi, L. Tensor Complementarity Problems—Part I: Basic Theory. J Optim Theory Appl 183, 1–23 (2019). https://doi.org/10.1007/s10957-019-01566-z

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