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

Exact and inexact Douglas-Rachford splitting methods are developed to solve the large-scale sparse absolute value equation (AVE) Ax - vertical bar x vertical bar = b, where A is an element of R-nxn and b is an element of 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 parallel to A(-1)parallel to < 1 and the solution set of the AVE is nonempty, the algorithms are globally and linearly convergent. When parallel to A(-1)parallel to = 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.