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On the cone eigenvalue complementarity problem for higher-order tensors

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Abstract

In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we give an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound on cone eigenvalues of tensors. Moreover, some new results concerning the bounds on the number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least, an implementable projection algorithm for solving TGEiCP is also developed for the problem under consideration. As an illustration of our theoretical results, preliminary computational results are reported.

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Acknowledgments

The authors would like to thank the two anonymous referees for their close readings and valuable recommendations, which help us improve the presentation of this paper significantly. The first two authors were supported by National Natural Science Foundation of China (NSFC) at Grant Nos. (11171083, 11301123) and the Zhejiang Provincial NSFC at Grant No. LZ14A010003. The third author was supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 502510, 502111, 501212 and 501913).

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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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  3. Liqun Qi
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Correspondence to Hongjin He.

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Ling, C., He, H. & Qi, L. On the cone eigenvalue complementarity problem for higher-order tensors. Comput Optim Appl 63, 143–168 (2016). https://doi.org/10.1007/s10589-015-9767-z

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