A backward-inertial Bregman forward-reflected backward algorithm for solving monotone inclusion problem

We study monotone inclusion problems in real Hilbert spaces
and propose two backward-inertial Bregman forward-reflected
backward splitting methods. Both algorithms modify the Malit
sky–Tam’s forward-reflected-backward scheme by replacing norm
distances and classical resolvents with Bregman distances and Breg
man resolvents, and incorporating inertial (momentum) step. Our
first method uses a constant stepsize, suitable when operator Lips
chitz constants are known; the second employs a self-adaptive step
sizethatdoesnotrequireLipschitzconstantstobeknown.Eachitera
tion performs only oneforward evaluation ofthe single-valuedoper
ator and one Bregman resolvent of the set-valued operator. Under
the assumptions that the single-valued operator is monotone and
Lipschitz continuous and the set-valued operator is maximal mono
tone, we prove weak convergence of both schemes to a solution
of the monotone inclusion problem. When, in addition, either the
single-valued operator or the set-valued operator is strongly mono
tone,weobtainstrongconvergence.Wefurtherestablishasublinear
rate of convergence. Using a notable Bregman distance, we perform
several numerical implementations and comparisons to support our
theoretical findings.