IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v205y2025i3d10.1007_s10957-025-02669-6.html
   My bibliography  Save this article

The Error and Perturbation Bounds of the General Absolute Value Equations

Author

Listed:
  • Cui-Xia Li

    (Yunnan Normal University)

  • Shi-Liang Wu

    (Yunnan Normal University
    Yunnan Normal University)

Abstract

To our knowledge, the error and perturbation bounds of the general absolute value equations (AVE) are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of these two AVEs: $$Ax-B|x|=b$$ A x - B | x | = b and $$Ax-|Bx|=b$$ A x - | B x | = b . Some useful error bounds and perturbation bounds of the above two AVEs are provided. Without limiting the matrix type, some computable estimates for the relevant upper bounds are given. By applying the absolute value equations, a new approach for some existing perturbation bounds of the linear complementarity problem (LCP) in (SIAM J. Optim., 18 (2007) 1250-1265) is provided. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.

Suggested Citation

  • Cui-Xia Li & Shi-Liang Wu, 2025. "The Error and Perturbation Bounds of the General Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 205(3), pages 1-21, June.
    • Handle: RePEc:spr:joptap:v:205:y:2025:i:3:d:10.1007_s10957-025-02669-6
    DOI: 10.1007/s10957-025-02669-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02669-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-025-02669-6?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Oleg Prokopyev, 2009. "On equivalent reformulations for absolute value equations," Computational Optimization and Applications, Springer, vol. 44(3), pages 363-372, December.
    2. Chao Zhang & Xiaojun Chen & Naihua Xiu, 2009. "Global error bounds for the extended vertical LCP," Computational Optimization and Applications, Springer, vol. 42(3), pages 335-352, April.
    3. Cuixia Li, 2022. "Sufficient Conditions for the Unique Solution of a New Class of Sylvester-Like Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 676-683, November.
    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. Shota Yamanaka & Nobuo Yamashita, 2018. "Duality of nonconvex optimization with positively homogeneous functions," Computational Optimization and Applications, Springer, vol. 71(2), pages 435-456, November.
    2. Shuhuang Xiang & Xiaojun Chen, 2011. "Computation of generalized differentials in nonlinear complementarity problems," Computational Optimization and Applications, Springer, vol. 50(2), pages 403-423, October.
    3. J. Y. Bello Cruz & O. P. Ferreira & L. F. Prudente, 2016. "On the global convergence of the inexact semi-smooth Newton method for absolute value equation," Computational Optimization and Applications, Springer, vol. 65(1), pages 93-108, September.
    4. Miao, Xin-He & Yang, Jiantao & Hu, Shenglong, 2015. "A generalized Newton method for absolute value equations associated with circular cones," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 155-168.
    5. Peng Wang & Yujing Zhang & Detong Zhu, 2025. "A New Finite-Difference Method for Nonlinear Absolute Value Equations," Mathematics, MDPI, vol. 13(5), pages 1-12, March.
    6. Olvi L. Mangasarian, 2014. "Absolute Value Equation Solution Via Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 870-876, June.
    7. Wang, Qianggang & Zhou, Yiyao & Fan, Bingxin & Liao, Jianquan & Huang, Tao & Zhang, Xuefei & Zou, Yao & Zhou, Niancheng, 2025. "Hierarchical optimal operation for bipolar DC distribution networks with remote residential communities," Applied Energy, Elsevier, vol. 378(PA).
    8. An Wang & Yang Cao & Jing-Xian Chen, 2019. "Modified Newton-Type Iteration Methods for Generalized Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 216-230, April.
    9. Yu, Dongmei & Zhang, Gehao & Ma, Tiange, 2025. "Neurodynamic approaches for solving absolute value equations and circuit implementation," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    10. Zhang, Jian-Jun, 2015. "The relaxed nonlinear PHSS-like iteration method for absolute value equations," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 266-274.
    11. Pedro Gajardo & Alberto Seeger, 2014. "Equilibrium problems involving the Lorentz cone," Journal of Global Optimization, Springer, vol. 58(2), pages 321-340, February.
    12. Hossein Moosaei & Saeed Ketabchi & Milan Hladík, 2021. "Optimal correction of the absolute value equations," Journal of Global Optimization, Springer, vol. 79(3), pages 645-667, March.
    13. Milan Hladík, 2018. "Bounds for the solutions of absolute value equations," Computational Optimization and Applications, Springer, vol. 69(1), pages 243-266, January.
    14. Cuixia Li, 2022. "Sufficient Conditions for the Unique Solution of a New Class of Sylvester-Like Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 676-683, November.

    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:spr:joptap:v:205:y:2025:i:3:d:10.1007_s10957-025-02669-6. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.