IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v514y2026ics0096300325005454.html

Nonnegative biquadratic tensors

Author

Listed:
  • Cui, Chunfeng
  • Qi, Liqun

Abstract

An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M+-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M+-eigenvalue is called an M++-eigenvalue. A nonnegative biquadratic tensor has at least one M+ eigenvalue, and the largest M+-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M+-eigenvalues are M++-eigenvalues. Although the M+-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M+-eigenvalue has a max-min characterization, while the smallest M+-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M+-eigenvalues is proposed. Numerical results are reported.

Suggested Citation

  • Cui, Chunfeng & Qi, Liqun, 2026. "Nonnegative biquadratic tensors," Applied Mathematics and Computation, Elsevier, vol. 514(C).
    • Handle: RePEc:eee:apmaco:v:514:y:2026:i:c:s0096300325005454
    DOI: 10.1016/j.amc.2025.129820
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300325005454
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2025.129820?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Chen, Yu & Hu, Zongqing & Hu, Jie & Shu, Lei, 2025. "Block structure-based covariance tensor decomposition for group identification in matrix variables," Statistics & Probability Letters, Elsevier, vol. 216(C).
    2. Liqun Qi & Chunfeng Cui & Yi Xu, 2025. "Quasi-Irreducibility of Nonnegative Biquadratic Tensors," Mathematics, MDPI, vol. 13(13), pages 1-10, June.
    3. Yuning Yang & Qingzhi Yang, 2012. "On solving biquadratic optimization via semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 53(3), pages 845-867, December.
    4. Jianxing Zhao & Pin Liu & Caili Sang, 2024. "Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor," Journal of Optimization Theory and Applications, Springer, vol. 200(3), pages 1131-1159, March.
    5. Gang Wang & Linxuan Sun & Lixia Liu, 2020. "M -Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors," Complexity, Hindawi, vol. 2020, pages 1-8, January.
    6. Ding, Weiyang & Liu, Jinjie & Qi, Liqun & Yan, Hong, 2020. "Elasticity M-tensors and the strong ellipticity condition," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shigui Li & Linzhang Lu & Xing Qiu & Zhen Chen & Delu Zeng, 2024. "Tighter bound estimation for efficient biquadratic optimization over unit spheres," Journal of Global Optimization, Springer, vol. 90(2), pages 323-353, October.
    2. Zhao, Jianxing, 2023. "Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    3. Li, Suhua & Li, Yaotang, 2021. "Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    4. Zhuolin Du & Chunyan Wang & Haibin Chen & Hong Yan, 2024. "An Efficient GIPM Algorithm for Computing the Smallest V-Singular Value of the Partially Symmetric Tensor," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1151-1167, June.
    5. Jianxing Zhao & Pin Liu & Caili Sang, 2024. "Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor," Journal of Optimization Theory and Applications, Springer, vol. 200(3), pages 1131-1159, March.
    6. Ying Zhang & Linxuan Sun & Gang Wang, 2020. "Sharp Bounds on the Minimum M -Eigenvalue of Elasticity M -Tensors," Mathematics, MDPI, vol. 8(2), pages 1-13, February.
    7. Ke Hou & Anthony Man-Cho So, 2014. "Hardness and Approximation Results for L p -Ball Constrained Homogeneous Polynomial Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1084-1108, November.
    8. Haibin Chen & Hongjin He & Yiju Wang & Guanglu Zhou, 2022. "An efficient alternating minimization method for fourth degree polynomial optimization," Journal of Global Optimization, Springer, vol. 82(1), pages 83-103, January.
    9. Liqun Qi & Chunfeng Cui & Yi Xu, 2025. "Quasi-Irreducibility of Nonnegative Biquadratic Tensors," Mathematics, MDPI, vol. 13(13), pages 1-10, June.
    10. Pengfei Huang & Qingzhi Yang & Yuning Yang, 2022. "Finding the global optimum of a class of quartic minimization problem," Computational Optimization and Applications, Springer, vol. 81(3), pages 923-954, April.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:514:y:2026:i:c:s0096300325005454. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.