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Structural Properties of Tensors and Complementarity Problems

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  • Yisheng Song

    (Henan Normal University)

  • Wei Mei

    (Henan Normal University)

Abstract

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.

Suggested Citation

  • Yisheng Song & Wei Mei, 2018. "Structural Properties of Tensors and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 289-305, February.
    • Handle: RePEc:spr:joptap:v:176:y:2018:i:2:d:10.1007_s10957-017-1212-2
    DOI: 10.1007/s10957-017-1212-2
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    References listed on IDEAS

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    1. Chen Ling & Hongjin He & Liqun Qi, 2016. "On the cone eigenvalue complementarity problem for higher-order tensors," Computational Optimization and Applications, Springer, vol. 63(1), pages 143-168, January.
    2. Xue-Li Bai & Zheng-Hai Huang & Yong Wang, 2016. "Global Uniqueness and Solvability for Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 72-84, July.
    3. Yisheng Song & Liqun Qi, 2016. "Eigenvalue analysis of constrained minimization problem for homogeneous polynomial," Journal of Global Optimization, Springer, vol. 64(3), pages 563-575, March.
    4. Zheng-Hai Huang & Liqun Qi, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    5. Chen Ling & Hongjin He & Liqun Qi, 2016. "Higher-degree eigenvalue complementarity problems for tensors," Computational Optimization and Applications, Springer, vol. 64(1), pages 149-176, May.
    6. 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.
    7. 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.
    8. Maolin Che & Liqun Qi & Yimin Wei, 2016. "Positive-Definite Tensors to Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 475-487, February.
    9. Hongchun Sun & Yiju Wang, 2013. "Further Discussion on the Error Bound for Generalized Linear Complementarity Problem over a Polyhedral Cone," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 93-107, October.
    10. Yisheng Song & Gaohang Yu, 2016. "Properties of Solution Set of Tensor Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 85-96, July.
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    Citations

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    Cited by:

    1. Wei Mei & Qingzhi Yang, 2020. "Properties of Structured Tensors and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 99-114, April.
    2. Lu-Bin Cui & Yu-Dong Fan & Yi-Sheng Song & Shi-Liang Wu, 2022. "The Existence and Uniqueness of Solution for Tensor Complementarity Problem and Related Systems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 321-334, January.
    3. Yisheng Song & Xudong Li, 2022. "Copositivity for a Class of Fourth-Order Symmetric Tensors Given by Scalar Dark Matter," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 334-346, October.
    4. Yang Xu & Guyan Ni & Mengshi Zhang, 2024. "Bounds of the Solution Set to the Polynomial Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 146-164, October.
    5. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 1-23, October.
    6. Xue-Li Bai & Zheng-Hai Huang & Xia Li, 2019. "Stability of Solutions and Continuity of Solution Maps of Tensor Complementarity Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-19, April.
    7. Vu Trung Hieu, 2019. "On the R0-Tensors and the Solution Map of Tensor Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 163-183, April.

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