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Tensor absolute value equations

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Abstract

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11671220, 11401331, 11771244 and 11271221), the Nature Science Foundation of Shandong Province (Grant Nos. ZR2015AQ013 and ZR2016AM29), and the Hong Kong Research Grant Council (Grant Nos. PolyU 501913, 15302114, 15300715 and 15301716). The authors thank the anonymous referees for their constructive comments and suggestions which led to a significantly improved version of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

    Shouqiang Du

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Liping Zhang & Chiyu Chen

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Liqun Qi

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  1. Shouqiang Du
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  2. Liping Zhang
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  3. Chiyu Chen
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  4. Liqun Qi
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Correspondence to Liping Zhang.

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Du, S., Zhang, L., Chen, C. et al. Tensor absolute value equations. Sci. China Math. 61, 1695–1710 (2018). https://doi.org/10.1007/s11425-017-9238-6

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  • DOI: https://doi.org/10.1007/s11425-017-9238-6

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