  Recently, we systematically studied the basic theory of Bregman circumcenters
in another paper. In this work, we aim to apply Bregman circumcenters to
optimization algorithms.
  Here, we propose the forward Bregman monotonicity which is a generalization
of the powerful Fej\'er monotonicity and show a weak convergence result of the
forward Bregman monotone sequence. We also naturally introduce the Bregman
circumcenter mappings associated with a finite set of operators. Then we
provide sufficient conditions for the sequence of iterations of the forward
Bregman circumcenter mapping to be forward Bregman monotone. Furthermore, we
prove that the sequence of iterations of the forward Bregman circumcenter
mapping weakly converges to a point in the intersection of the fixed point sets
of relevant operators, which reduces to the known weak convergence result of
the circumcentered method under the Euclidean distance. In addition, particular
examples are provided to illustrate the Bregman isometry and Browder's
demiclosedness principle, and our convergence result.
