  We demonstrate the relevance of an algorithm called generalized iterative
scaling (GIS) or simultaneous multiplicative algebraic reconstruction technique
(SMART) and its rescaled block-iterative version (RBI-SMART) in the field of
optimal transport (OT). Many OT problems can be tackled through the use of
entropic regularization by solving the Schr\"odinger problem, which is an
information projection problem, that is, with respect to the Kullback--Leibler
divergence. Here we consider problems that have several affine constraints. It
is well-known that cyclic information projections onto the individual affine
sets converge to the solution. In practice, however, even these individual
projections are not explicitly available in general. In this paper, we exchange
them for one GIS iteration. If this is done for every affine set, we obtain
RBI-SMART. We provide a convergence proof using an interpretation of these
iterations as two-step affine projections in an equivalent problem. This is
done in a slightly more general setting than RBI-SMART, since we use a mix of
explicitly known information projections and GIS iterations. We proceed to
specialize this algorithm to several OT applications. First, we find the
measure that minimizes the regularized OT divergence to a given measure under
moment constraints. Second and third, the proposed framework yields an
algorithm for solving a regularized martingale OT problem, as well as a relaxed
version of the barycentric weak OT problem. Finally, we show an approach from
the literature for unbalanced OT problems.
