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Generalized tensor equations with leading structured tensors

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  • Yan, Weijie
  • Ling, Chen
  • Ling, Liyun
  • He, Hongjin

Abstract

Systems of tensor equations (TEs) have received much considerable attention in the recent literature. In this paper, we consider a class of generalized tensor equations (GTEs). An important difference between GTEs and TEs is that GTEs can be regarded as a system of non-homogenous polynomial equations, whereas TEs is a homogenous one. Such a difference usually make the theoretical and algorithmic results tailored for TEs not necessarily applicable to GTEs. To study properties of the solution set of GTEs, we first introduce a new class of so-named Z+-tensors, which includes the set of all P-tensors as a proper subset. With the help of degree theory, we prove that the system of GTEs with a leading coefficient Z+-tensor has at least one solution for any right-hand side vector. Moreover, we study the local error bounds under some appropriate conditions. Finally, we employ a Levenberg-Marquardt algorithm to find a solution to GTEs and report some preliminary numerical results.

Suggested Citation

  • Yan, Weijie & Ling, Chen & Ling, Liyun & He, Hongjin, 2019. "Generalized tensor equations with leading structured tensors," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 311-324.
    • Handle: RePEc:eee:apmaco:v:361:y:2019:i:c:p:311-324
    DOI: 10.1016/j.amc.2019.05.042
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    References listed on IDEAS

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    1. Yong Wang & Zheng-Hai Huang & Liqun Qi, 2018. "Global Uniqueness and Solvability of Tensor Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 137-152, April.
    2. Shui-Lian Xie & Dong-Hui Li & Hong-Ru Xu, 2017. "An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 119-136, October.
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    5. Yisheng Song & Liqun Qi, 2015. "Properties of Some Classes of Structured Tensors," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 854-873, June.
    6. Yisheng Song & Liqun Qi, 2016. "Tensor Complementarity Problem and Semi-positive Tensors," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1069-1078, June.
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    Cited by:

    1. Dong-Hui Li & Jie-Feng Xu & Hong-Bo Guan, 2021. "Newton’s Method for M-Tensor Equations," Journal of Optimization Theory and Applications, Springer, vol. 190(2), pages 628-649, August.

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