Milan Hladík and Lenka Ptáčková. Absolute value equations with interval uncertainty. Soft Comput., 29(7):3705–3718, April 2025.
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We consider the (generalized) absolute value equations with uncertain entries. In particular, we assume that the entries come from given intervals. For such interval-valued problem, we study the fundamental properties regarding solutions and solvability. First, we address the issue of the unique solvability of each realization. This is a hard problem, but it can be characterized by means of a regularity of a set of matrices or by a nonlinear system of inequalities; in contrast, the nonnegative unique solvability is polynomially decidable. Second, we focus on the overall solution set. We provide a closed-form characterization and inspect its topological properties such as boundedness. Since the solution set is hard to deal with, we derive a formula for an outer approximation and analyze the situation when the approximation is tight. Eventually, we investigate the convex hull of the solution set; we present an explicit formula that yields the convex hull under mild assumptions.
@article{HlaPta2025,
author = "Milan Hlad\'{\i}k and Lenka Pt\'{a}\v{c}kov\'{a}",
title = "Absolute value equations with interval uncertainty",
journal = "Soft Comput.",
fjournal = "Soft Computing",
volume = "29",
number = "7",
pages = "3705-3718",
month = "April",
year = "2025",
doi = "10.1007/s00500-025-10655-3",
issn = "1432-7643",
issnonline = "1433-7479",
url = "https://doi.org/10.1007/s00500-025-10655-3",
bib2html_dl_html = "https://link.springer.com/article/10.1007/s00500-025-10655-3",
bib2html_dl_pdf = "https://rdcu.be/etCSj",
abstract = "We consider the (generalized) absolute value equations with uncertain entries. In particular, we assume that the entries come from given intervals. For such interval-valued problem, we study the fundamental properties regarding solutions and solvability. First, we address the issue of the unique solvability of each realization. This is a hard problem, but it can be characterized by means of a regularity of a set of matrices or by a nonlinear system of inequalities; in contrast, the nonnegative unique solvability is polynomially decidable. Second, we focus on the overall solution set. We provide a closed-form characterization and inspect its topological properties such as boundedness. Since the solution set is hard to deal with, we derive a formula for an outer approximation and analyze the situation when the approximation is tight. Eventually, we investigate the convex hull of the solution set; we present an explicit formula that yields the convex hull under mild assumptions.",
keywords = "Absolute value equations; Interval matrix; Interval analysis; NP-hardness; Linear complementarity problem",
}
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