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

Abstract

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