An Optimization Problem for Mass Transportation with Congested Dynamics

Buttazzo, G.; Jimenez, C.; Oudet, E. · 2009 · OpenAlex-citations

DOI: 10.1137/07070543x

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Summary

This paper addresses the optimization of mass transportation in scenarios involving congested dynamics, such as traffic flow or crowd movement where high density impedes motion. While classical Monge-Kantorovich theory and the dynamic formulation by Brenier assume particles move along straight lines or geodesics to minimize kinetic energy, these models fail to account for congestion effects that increase costs and alter trajectories. The authors generalize the framework to include cost functions with superlinear growth relative to density, thereby penalizing high concentrations and modeling realistic constraints like maximum occupancy limits. The study formulates the problem as minimizing a functional $\Psi(\sigma)$ over vectorial measures $\sigma = (\rho, E)$ representing mass density and flux, subject to a continuity equation constraint linking initial and final distributions. The authors establish existence and uniqueness results for the primal problem under specific convexity conditions and derive the corresponding dual formulation. They analyze various cost structures, including the standard Wasserstein distance, a model penalizing high density via a quadratic term, and a model enforcing a hard upper bound on density. The theoretical analysis includes detailed derivations of primal-dual optimality conditions, utilizing relaxed dual spaces and tangent space projections to handle non-smooth dual variables. To solve these problems numerically, the authors adapt the augmented Lagrangian scheme introduced by Benamou and Brenier. The algorithm iteratively updates the dual potential by solving a perturbed Laplace equation and updates the primal variables through pointwise minimization of the augmented Lagrangian. The authors demonstrate the method’s efficacy through numerical experiments on non-convex domains containing obstacles. In these simulations, mass is transported from a Gaussian source to a target distribution. The results show that when congestion terms are included, the mass density spreads out to avoid high-density regions, deviating from the concentrated geodesic paths observed in the unconstrained Wasserstein case. This confirms that the proposed framework effectively captures congestion-induced dispersion and adherence to density constraints. The significance of this work lies in providing a rigorous mathematical foundation and computational tool for modeling mass transport with congestion. By extending the dynamic optimal transport framework to include density-dependent costs, the paper bridges the gap between abstract transport theory and practical applications in urban planning, crowd dynamics, and traffic engineering. The results demonstrate that congestion fundamentally alters optimal transport paths, necessitating models that account for density penalties rather than relying solely on geometric distances.

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• traffic density