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Higher-degree eigenvalue complementarity problems for tensors

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Abstract

In this paper, we introduce a unified framework of Tensor Higher-Degree Eigenvalue Complementarity Problem (THDEiCP), which goes beyond the framework of the typical Quadratic Eigenvalue Complementarity Problem for matrices. First, we study some topological properties of higher-degree cone eigenvalues of tensors. Based upon the symmetry assumptions on the underlying tensors, we then reformulate THDEiCP as a weakly coupled homogeneous polynomial optimization problem, which might be greatly helpful for designing implementable algorithms to solve the problem under consideration numerically. As more general theoretical results, we present the results concerning existence of solutions of THDEiCP without symmetry conditions. Finally, we propose an easily implementable algorithm to solve THDEiCP, and report some computational results.

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Acknowledgments

The authors would like to thank the two anonymous referees for their careful reading and valuable comments, which significantly improved the presentation of this paper. The first two authors would like to thank Professor Deren Han for his comments on the numerical algorithm. This research of the first two authors was supported in part by National Natural Science Foundation of China (11171083, 11301123, 11571087) and the Zhejiang Provincial NSF (LZ14A010003). The third author was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 502111, 501212, 501913, and 15302114).

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Authors and Affiliations

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

    Chen Ling & Hongjin He

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

    Liqun Qi

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  1. Chen Ling
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  2. Hongjin He
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Correspondence to Hongjin He.

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Ling, C., He, H. & Qi, L. Higher-degree eigenvalue complementarity problems for tensors. Comput Optim Appl 64, 149–176 (2016). https://doi.org/10.1007/s10589-015-9805-x

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