A class of maximum-based iteration methods for the generalized absolute value equation
Shiliang Wu, Deren Han, Cuixia Li
TL;DR
The paper addresses solving the generalized absolute value equation $A x - B|x| = b$ by introducing a maximum-based iteration framework that uses $|x|=2\max\{0,x\}-x$ to avoid direct absolute-value evaluations and exploit $B$-information. It presents Method 3, with update $x^{k+1}=(A+B+\Omega)^{-1}(\Omega x^{k}+2B\max\{0,x^{k}\}+b)$, requiring $(A+B+\Omega)$ to be invertible. A global convergence criterion is established: $\rho(f_{\Omega}+g_{\Omega})<1$ where $f_{\Omega}=|(A+B+\Omega)^{-1}\Omega|$ and $g_{\Omega}=2|(A+B+\Omega)^{-1}B|$, and the section analyzes special cases when $A+B$ is SPD or an $H_{+}$-matrix, showing when the problem reduces to linear-system solves. Numerical experiments on two GAVE instances demonstrate that Method 3 often requires fewer iterations and less CPU time than existing methods (including the modified Newton method and GMRES) while achieving smaller residuals, validating the method’s effectiveness and practical value.
Abstract
In this paper, by using $|x|=2\max\{0,x\}-x$, a class of maximum-based iteration methods is established to solve the generalized absolute value equation $Ax-B|x|=b$. Some convergence conditions of the proposed method are presented. By some numerical experiments, the effectiveness and feasibility of the proposed method are confirmed.