  In this paper, we propose and study the iteration complexity of an inexact
Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng's
forward-backward (F-B) splitting method for solving two-operator and
four-operator monotone inclusions, respectively. The former method (although
based on a slightly different mechanism of iteration) is motivated by the
recent work of J. Eckstein and W. Yao, in which an inexact DRS method is
derived from a special instance of the hybrid proximal extragradient (HPE)
method of Solodov and Svaiter, while the latter one combines the proposed
inexact DRS method (used as an outer iteration) with a Tseng's F-B splitting
type method (used as an inner iteration) for solving the corresponding
subproblems. We prove iteration complexity bounds for both algorithms in the
pointwise (non-ergodic) as well as in the ergodic sense by showing that they
admit two different iterations: one that can be embedded into the HPE method,
for which the iteration complexity is known since the work of Monteiro and
Svaiter, and another one which demands a separate analysis. Finally, we perform
simple numerical experiments %on three-operator and four-operator monotone
inclusions to show the performance of the proposed methods when compared with
other existing algorithms.
