Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium
Tigran Bakaryan, Christoph Aoun, Ricardo de Lima Ribeiro, Naira Hovakimyan, Diogo Gomes
TL;DR
The work addresses multi-population Wardrop equilibria on generalized networks with multiple entry/exit points and heterogeneous costs. It proves existence and, under strict monotonicity, uniqueness, and reformulates the problem as a distributed optimization solvable by a Hessian Riemannian flow (HRF) that preserves feasibility and converges globally. The HRF approach is applied to scenarios spanning uniform, heterogeneous, and emissions-aware traffic, demonstrating computational efficiency (often sub-second convergence) and the ability to distribute traffic to minimize emissions. This yields a scalable framework for realistic traffic management and environmental optimization in urban networks.
Abstract
In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population Wardrop equilibrium, defined through variational inequalities. We rigorously analyze the existence and uniqueness of the Wardrop equilibrium. Furthermore, we introduce an efficient numerical method to find the solution. In particular, we reformulate the equilibrium problem as a distributed optimization problem over subgraphs and introduce a novel Hessian Riemannian flow method, a Riemannian-manifold-projected Hessian flow, to efficiently compute a solution. Finally, we demonstrate the effectiveness of our approach through examples in urban traffic management, including routing for diverse vehicle types and strategies for minimizing emissions in congested environments.