  In this work we develop a numerical method for solving a type of convex
graph-structured tensor optimization problems. This type of problems, which can
be seen as a generalization of multi-marginal optimal transport problems with
graph-structured costs, appear in many applications. Examples are unbalanced
optimal transport and multi-species potential mean field games, where the
latter is a class of nonlinear density control problems. The method we develop
is based on coordinate ascent in a Lagrangian dual, and under mild assumptions
we prove that the algorithm converges globally. Moreover, under a set of
stricter assumptions, the algorithm converges R-linearly. To perform the
coordinate ascent steps one has to compute projections of the tensor, and doing
so by brute force is in general not computationally feasible. Nevertheless, for
certain graph structures it is possible to derive efficient methods for
computing these projections, and here we specifically consider the graph
structure that occurs in multi-species potential mean field games. We also
illustrate the methodology on a numerical example from this problem class.
