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Method of alternating projections for the general absolute value equation

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Abstract

A novel approach for solving the general absolute value equation \(Ax+B|x| = c\) where \(A,B\in \textrm{I}\! \textrm{R}^{m\times n}\) and \(c\in \textrm{I}\! \textrm{R}^m\) is presented. We reformulate the equation as a nonconvex feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on A and B. Furthermore, we prove local linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with \(m\ne n\), both theoretically and numerically.

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Acknowledgements

J. H. Alcantara and J.-S. Chen’s research is supported by Ministry of Science and Technology, Taiwan. M. K. Tam is supported in part by DE200100063 from the Australian Research Council. This work was conducted while J. H. Alcantara was a postdoctoral fellow at National Taiwan Normal University. The authors would like to thank the anonymous referee for the valuable feedback and comments.

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Authors and Affiliations

  1. Institute of Statistical Sciences, Academia Sinica, Taipei, 11529, Taiwan

    Jan Harold Alcantara

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    Jein-Shan Chen

  3. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Matthew K. Tam

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  1. Jan Harold Alcantara
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  2. Jein-Shan Chen
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  3. Matthew K. Tam
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Correspondence to Jein-Shan Chen.

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Alcantara, J.H., Chen, JS. & Tam, M.K. Method of alternating projections for the general absolute value equation. J. Fixed Point Theory Appl. 25, 39 (2023). https://doi.org/10.1007/s11784-022-01026-8

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