Milan Hladík. Overconstrained and underconstrained systems of absolute value equations. SIAM J. Matrix Anal. Appl., 47(1):244–264, March 2026.
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Absolute value equations are nonlinear systems of the form Ax+B|x|=b, and they have seen rapid development in recent years. So far, the research has been focused primarily to the square case. In this paper, we aim to shift this focus and draw attention to overdetermined and underdetermined systems. For these rectangular cases, we address fundamental solvability issues. In particular, for overdetermined systems, we fully characterize cases where at most one solution exists. Similarly, for underdetermined systems, we present various conditions for the existence of at least one solution. The solvability of these systems is closely related to the full rank properties of certain sets of matrices. That is why we also investigate the full rank problems and derive different characterization conditions. Since absolute value equations are challenging to handle and many related problems are NP-hard, we examine rank-one (or more generally, fixed-rank) cases and demonstrate that certain questions can be answered efficiently. Finally, we pose several open problems, which are both easy to formulate and fundamental to solvability analysis.
@article{Hla2026a,
author = "Milan Hlad\'{\i}k",
title = "Overconstrained and underconstrained systems of absolute value equations",
journal = "SIAM J. Matrix Anal. Appl.",
fjournal = "SIAM Journal on Matrix Analysis and Applications",
volume = "47",
number = "1",
pages = "244-264",
month = "March",
year = "2026",
doi = "10.1137/25M1744563",
issn = "0895-4798",
issnweb = "1095-7162",
url = "https://epubs.siam.org/doi/full/10.1137/25M1744563",
bib2html_dl_html = "https://doi.org/10.1137/25M1744563",
bib2html_dl_pdf = "https://epubs.siam.org/doi/epdf/10.1137/25M1744563",
abstract = "Absolute value equations are nonlinear systems of the form Ax+B|x|=b, and they have seen rapid development in recent years. So far, the research has been focused primarily to the square case. In this paper, we aim to shift this focus and draw attention to overdetermined and underdetermined systems. For these rectangular cases, we address fundamental solvability issues. In particular, for overdetermined systems, we fully characterize cases where at most one solution exists. Similarly, for underdetermined systems, we present various conditions for the existence of at least one solution. The solvability of these systems is closely related to the full rank properties of certain sets of matrices. That is why we also investigate the full rank problems and derive different characterization conditions. Since absolute value equations are challenging to handle and many related problems are NP-hard, we examine rank-one (or more generally, fixed-rank) cases and demonstrate that certain questions can be answered efficiently. Finally, we pose several open problems, which are both easy to formulate and fundamental to solvability analysis.",
keywords = "Absolute value equations; Linear complementarity problem; NP-hardness; Overdetermined system; Underedetermined system; Interval matrix; Interval analysis",
}
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