  Entropy regularized optimal transport and its multi-marginal generalization
have attracted increasing attention in various applications, in particular due
to efficient Sinkhorn-like algorithms for computing optimal transport plans.
However, it is often desirable that the marginals of the optimal transport plan
do not match the given measures exactly, which led to the introduction of the
so-called unbalanced optimal transport. Since unbalanced methods were not
examined for the multi-marginal setting so far, we address this topic in the
present paper. More precisely, we introduce the unbalanced multi-marginal
optimal transport problem and its dual, and show that a unique optimal
transport plan exists under mild assumptions. Further, we generalize the
Sinkhorn algorithm for regularized unbalanced optimal transport to the
multi-marginal setting and prove its convergence. If the cost function
decouples according to a tree, the iterates can be computed efficiently. At the
end, we discuss three applications of our framework, namely two barycenter
problems and a transfer operator approach, where we establish a relation
between the barycenter problem and the multi-marginal optimal transport with an
appropriate tree-structured cost function.

Optimization and Control