GRAPH-STRUCTURED TENSOR OPTIMIZATION FOR NONLINEAR DENSITY CONTROL AND MEAN FIELD GAMES

In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problem. This type of problem, which can be seen as a generalization of multimarginal optimal transport problems with graph-structured costs, appears in many applications. Examples are unbalanced optimal transport and multispecies 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 multispecies potential mean field games. We also illustrate the methodology on a numerical example from this problem class.