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 − |x| = b, where A ∈ Rⁿ×ⁿ and b ∈ Rⁿ. 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 ⃦ ≤ 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.