  We study a general convex optimization problem, which covers various classic
problems in different areas and particularly includes many optimal transport
related problems arising in recent years. To solve this problem, we revisit the
classic Bregman proximal point algorithm (BPPA) and introduce a new inexact
stopping condition for solving the subproblems, which can circumvent the
underlying feasibility difficulty often appearing in existing inexact
conditions when the problem has a complex feasible set. Our inexact condition
also covers several existing inexact conditions as special cases and hence
makes our inexact BPPA (iBPPA) more flexible to fit different scenarios in
practice. Moreover, inspired by Nesterov's acceleration technique, we develop
an inertial variant of our iBPPA, denoted by V-iBPPA, and establish the
iteration complexity of $O(1/k^{\lambda})$, where $\lambda\geq1$ is a
quadrangle scaling exponent of the kernel function. In particular, when the
proximal parameter is a constant and the kernel function is strongly convex
with Lipschitz continuous gradient (hence $\lambda=2$), our V-iBPPA achieves a
faster rate of $O(1/k^2)$ just as existing accelerated inexact proximal point
algorithms. Some preliminary numerical experiments for solving the standard OT
problem are conducted to show the convergence behaviors of our iBPPA and
V-iBPPA under different inexactness settings. The experiments also empirically
verify the potential of our V-iBPPA on improving the convergence speed.
