Exact and inexact Douglas–Rachford splitting methods for solving large-scale sparse absolute value equations

Abstract Exact and inexact Douglas–Rachford splitting methods are developed to solve the large-scale sparse absolute value equation (AVE) $Ax - |x| =b$, where $A\in \mathbb {R}^{n\times n}$ $b\in {R}^n$. The method adopts a relative error tolerance and, therefore, in inner iterative processes, LSQR is employed find qualified approximate solution of each subproblem, resulting lower cost for iteration. When $\|A^{-1}\|\le 1$ set AVE nonempty, algorithms globally linearly convergent. $\|A^{-1}\|= empty, sequence generated by exact algorithm diverges infinity on trivial example. Numerical examples presented demonstrate viability robustness proposed methods.