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A General Preconditioner for Tensor Complementarity Problems

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Abstract

Preconditioning techniques have been focused on multi-linear systems or tensor equations. However, to our knowledge, there is no relevant research on tensor complementarity problems. In this paper, we present a general preconditioner P for solving the tensor complementarity problem (TCP) involved with an \({\mathcal {L}}\)-tensor \({\mathcal {A}}\) and a vector q, and we prove that the TCP\(({\mathcal {A}},q)\) is equivalent to the TCP\((P{\mathcal {A}},Pq)\) under the assumption that \(P{\mathcal {A}}\) is an \({\mathcal {L}}\)-tensor. Based on this equivalence, a preconditioned fixed point iteration method is proposed for solving the tensor complementarity problem and its convergence analysis is given. For actual computations, we provide a concrete choice for the preconditioner P associated with a parameter satisfying above-mentioned hypothesis. In addition, it is proved theoretically that the convergence rate of the fixed point method with the chosen preconditioner P is at least as fast as that of the corresponding method without preprocessing. Meanwhile, we also obtain the monotony of the parameter on the performance of the preconditioned iterative method. Lastly, numerical examples are used to demonstrate the theoretical results.

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Acknowledgements

The author would like to thank the editor and anonymous referees for their valuable comments and suggestions, which greatly improve the quality of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, Guangdong, People’s Republic of China

    Ping-Fan Dai

  2. Research and Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen, 518057, Guangdong, People’s Republic of China

    Jianchao Bai

  3. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Jianchao Bai

  4. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Jicheng Li

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  1. Ping-Fan Dai
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Correspondence to Jianchao Bai.

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This work is partly supported by the Fund provided by the Department of Education of Guangdong Province (2023KTSCX080), the Professor and Doctor Start Program of Hanshan Normal University (QD202325), the NSF of China (12171384), the Guangdong Basic and Applied Basic Research Foundation (2023A1515012405) and the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSQ001).

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Dai, PF., Bai, J. & Li, J. A General Preconditioner for Tensor Complementarity Problems. J Sci Comput 98, 3 (2024). https://doi.org/10.1007/s10915-023-02391-3

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  • DOI: https://doi.org/10.1007/s10915-023-02391-3

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