Milan Hladík and Hossein Moosaei. Some notes on the solvability conditions for absolute value equations. Optim. Lett., 17(1):211–218, January 2023.
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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.
@article{HlaMoo2023a,
author = "Milan Hlad\'{\i}k and Hossein Moosaei",
title = "Some notes on the solvability conditions for absolute value equations",
journal = "Optim. Lett.",
fjournal = "Optimization Letters",
volume = "17",
number = "1",
month = "January",
pages = "211-218",
year = "2023",
doi = "10.1007/s11590-022-01900-x",
issn = "1862-4480",
url = "https://link.springer.com/article/10.1007/s11590-022-01900-x",
bib2html_dl_html = "https://doi.org/10.1007/s11590-022-01900-x",
bib2html_dl_pdf = "https://rdcu.be/c2NK5",
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.",
keywords = "Absolute value equations; Interval matrix; Linear complementarity problem",
}
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