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Some notes on the solvability conditions for absolute value equations

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Abstract

Motivated by recently published papers, we discuss the problem of solving absolute value equations of the form \(Ax+|x|=b\). Specifically, we show the equivalence of two sufficient conditions for the unsolvability of such equations which are based on linear programming. Furthermore, we prove that two generalizations of sufficient conditions for unique solvability do not extend the class at all, that is, they are equivalent to the original formulations. The conditions based on M-matrices and H-matrices also turn out to be special cases of the known statements. Eventually, we also point out two incorrect statements, and for one of them, we propose some corrected reformulations.

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Acknowledgements

The authors were supported by the Czech Science Foundation under Grant P403-22-11117S. The work of H. Moosaei was also supported by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, Malostranske nam. 25, 118 00, Prague, Czech Republic

    Milan Hladík & Hossein Moosaei

  2. Faculty of Science, Department of Mathematics, University of Bojnord, Bojnord, Iran

    Hossein Moosaei

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  1. Milan Hladík
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  2. Hossein Moosaei
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Correspondence to Milan Hladík.

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Hladík, M., Moosaei, H. Some notes on the solvability conditions for absolute value equations. Optim Lett 17, 211–218 (2023). https://doi.org/10.1007/s11590-022-01900-x

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  • DOI: https://doi.org/10.1007/s11590-022-01900-x

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