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Acceptable Solutions and Backward Errors for Tensor Complementarity Problems

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Abstract

Backward error analysis reveals the numerical stability of algorithms and provides elaborate stopping criteria for iterative methods. Compared with numerical linear algebra problems, the backward error analysis for optimization problems is more rarely conducted in the literature. This paper is devoted to the backward error analysis for several generalizations of tensor complementarity problems. We first present sufficient and necessary conditions for the acceptable solutions for the extended tensor complementarity problem, the vertical tensor complementarity problem, and an extended form of tensor complementarity problem. Next, the backward errors for tensor complementarity problem are also proposed, which can be used to verify the stability of the tensor complementarity problem algorithms. Finally, some numerical examples are reported to illustrate the proposed backward errors for tensor complementarity problems.

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Acknowledgements

The authors thank the editor and the anonymous referees for numerous insightful comments, which have greatly improved this paper. S. Du is supported by the National Natural Science Foundation of China under grant 11671220. W. Ding is supported by the National Natural Science Foundation of China under grant 11801479 and the Hong Kong Research Grants Council under grant 12301619. Y. Wei is supported by the National Natural Science Foundation of China under grant 11771099 and the Innovation Program of Shanghai Municipal Education Committee.

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Authors and Affiliations

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

    Shouqiang Du

  2. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, 200433, China

    Weiyang Ding

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Yimin Wei

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  1. Shouqiang Du
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  2. Weiyang Ding
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  3. Yimin Wei
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Correspondence to Yimin Wei.

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Communicated by Guoyin Li.

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Du, S., Ding, W. & Wei, Y. Acceptable Solutions and Backward Errors for Tensor Complementarity Problems. J Optim Theory Appl 188, 260–276 (2021). https://doi.org/10.1007/s10957-020-01774-y

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