Speaker
Description
We investigate the structural properties of the solution set of absolute value equations (AVE) of the form $Ax - |x| = b$. Extending the seminal work of Hladík (SIAM J. Matrix Anal. Appl., 2023), we address his open questions originally posed for $Ax + |x| = b$ and establish analogous results for the alternative form considered here. Using the equivalence between AVE and the linear complementarity problem (LCP), we derive new results on the convexity and solvability of the AVE under various matrix classes, including positive semidefinite, Metzler, and $Z$-matrices. Further, we present sufficient conditions guaranteeing unique and nonnegative solutions for $b \ge 0$ or $b \le 0$, and provide counterexamples showing that several properties valid for $Ax + |x| = b$ fail to extend to $Ax - |x| = b$. These findings contribute to the theoretical understanding of AVEs and enrich the connection between matrix theory and complementarity problems.