Graph-structured tensor optimization for nonlinear density control and mean field games

TL;DR

This work tackles convex graph-structured tensor optimization, a generalization of graph-structured multi-marginal optimal transport that arises in unbalanced OT and multi-species potential mean field games. It develops a dual-coordinate-ascent ( Sinkhorn-type ) algorithm that exploits a graph decomposition of marginals and bimarginals, with entropy regularization ensuring tractable updates. The paper proves global convergence and, under stronger assumptions, R-linear convergence, and provides a reformulation yielding existence and uniqueness guarantees, a no-duality-gap dual, and extensions to multiple costs per marginal. A key contribution is the specialized algorithm for multi-species density control, including a detailed discretization to a path-space tensor and a 2D numerical example demonstrating density evolution and inter-species interactions. The framework offers a scalable approach to large-scale network flow and density-control problems with heterogeneous agents, enabling efficient computation through graph-aware projections and coordinated dual updates.

Abstract

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.

Figures (2)

• Figure 1: Illustration of the graph ${\mathcal{G}}$ for the multi-species density optimal control problem. Grey circles correspond to known densities, and white circles correspond to densities which are to be optimized over.
• Figure 2: Figures describing the setup in the numerical example in Section \ref{['sec:2D_example']}. (a) Target densities $\tilde{\mu}_1$ (left) and $\tilde{\mu}_2$ (right) for the total density at time points $j=19$ and $j=39$, respectively. (b) Illustration of species-dependent cost and constraint: left plot shows the linear cost $c_3$ for species $3$, where blue means cost 0 and yellow means a cost of $390\Delta x\Delta t$. The right plots shows the target distributions $\tilde{\nu}$ for species $4$. (c) The capacity constraint $\kappa_j$ at the different time points $j$: blue means zero capacity (obstacle) while yellow means infinite capacity. (d) The optimal solution, illustrated as time evolution of total density and densities of the individual species.

Theorems & Definitions (30)

• Remark 3.1
• Remark 3.2
• Lemma 3.4
• Proof 1
• Remark 3.5
• Theorem 3.6
• Proof 2
• Corollary 3.8
• Proof 3
• Lemma 3.10