Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium

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 co st structures. We tackle this problem by considering the multi - population Wardrop equilibrium, defined through variation al inequalities. We rigorously analyze the existence and uniq ueness of the Wardrop equilibrium. Furthermore, we introduce an efficient numerical method to find the solution. In particula r, 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. Fi - nally, we demonstrate the effectiveness of our approach thr ough examples in urban traffic management, including routing for diverse vehicle types and strategies for minimizing emissi ons in congested environments. In traffic management, each driver--whether operating a car, SUV, or truck--selects the route they perceive to be the shortest.