  The Bregman proximal gradient method (BPGM), which uses the Bregman distance
as a proximity measure in the iterative scheme, has recently been re-developed
for minimizing convex composite problems \textit{without} the global Lipschitz
gradient continuity assumption. This makes the BPGM appealing for a wide range
of applications, and hence it has received growing attention in recent years.
However, most existing convergence results are only obtained under the
assumption that the involved subproblems are solved \textit{exactly}, which is
not realistic in many applications. For the BPGM to be implementable and
practical, in this paper, we develop inexact versions of the BPGM by employing
either an absolute-type stopping criterion or a relative-type stopping
criterion solving the subproblems. The iteration complexity of
$\mathcal{O}(1/k)$ and the convergence of the sequence are also established for
our iBPGM under some conditions. Moreover, we develop an inertial variant of
our iBPGM (denoted by v-iBPGM) and establish the iteration complexity of
$\mathcal{O}(1/k^{\gamma})$, where $\gamma\geq1$ is a restricted relative
smoothness exponent. When the smooth part in the objective has a Lipschitz
continuous gradient and the kernel function is strongly convex, we have
$\gamma=2$ and thus the v-iBPGM improves the iteration complexity of the iBPGM
from $\mathcal{O}(1/k)$ to $\mathcal{O}(1/k^2)$, in accordance with the
existing results on the exact accelerated BPGM. Finally, some preliminary
numerical experiments for solving the discrete quadratic regularized optimal
transport problem are conducted to illustrate the convergence behaviors of our
iBPGM and v-iBPGM under different inexactness settings.
