Bounds for the solutions of absolute value equations

In the recent years, there has been an intensive research of absolute value equations Ax-b=B|x|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-b=B|x|$$\end{document}. Various methods were developed, but less attention has been paid to approximating or bounding the solutions. We start filling this gap by proposing several outer approximations of the solution set. We present conditions for unsolvability and for existence of exponentially many solutions, too, and compare them with the known conditions. Eventually, we carried out numerical experiments to compare the methods with respect to computational time and quality of estimation. This helps in identifying the cases, in which the bounds are tight enough to determine the signs of the solution, and therefore also the solution itself.