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The Error and Perturbation Bounds of the General Absolute Value Equations

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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: \(Ax-B|x|=b\) and \(Ax-|Bx|=b\). 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.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon request.

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Acknowledgements

The authors are very much indebted to two anonymous referees for their constructive suggestions and helpful comments, which led to significant improvement of the original manuscript of this paper.

Funding

This research was supported by National Natural Science Foundation of China (No.11961082), Cross-integration Innovation team of modern Applied Mathematics and Life Sciences in Yunnan Province, China (No.202405AS350003), Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (202302AN360007), Basic Research Projects of Key Scientific Research Projects Plan in Henan Higher Education Institutions (No. 25zx004).

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Authors and Affiliations

  1. School of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, P.R. China

    Cui-Xia Li & Shi-Liang Wu

  2. Yunnan Key Laboratory of Modern Analytical Mathematics and Applications, Yunnan Normal University, Kunming, 650500, Yunnan, P.R. China

    Shi-Liang Wu

Authors
  1. Cui-Xia Li
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  2. Shi-Liang Wu
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Correspondence to Shi-Liang Wu.

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Communicated by Nobuo Yamashita.

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Li, CX., Wu, SL. The Error and Perturbation Bounds of the General Absolute Value Equations. J Optim Theory Appl 205, 53 (2025). https://doi.org/10.1007/s10957-025-02669-6

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