Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and $\mathcal{W}$-property. Based on the unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP)

On the unique solvability and numerical study of absolute value equations

The aim of this paper is twofold. Firstly, we consider the unique solvability of absolute value equations (AVE), $Ax-B\vert x\vert =b$, when the condition $\Vert A^{-1}\Vert <\frac{1}{\left\Vert B\right\Vert }$ holds. This is a generalization of an earlier result by Mangasarian and Meyer for the special case where $B=I$. Secondly, a generalized Newton method for solving the AVE is proposed. We show under the condition $\Vert A^{-1}\Vert <\frac{1}{4\Vert B\Vert }$, that the algorithm converges linearly global to the unique solution of the AVE. Numerical results are reported to show the efficiency of the proposed method and to compare with an available method