Distributed Hessian–Riemannian flow for multi-population wardrop equilibrium

In this paper, we studied a multi-population Wardrop equilibrium on generalized directed graphs with heterogeneous costs. Using an edge-flow representation, we formulated the equilibrium as a variational inequality and provided sufficient conditions for existence and uniqueness under compactness, continuity, and monotonicity assumptions. We showed that the equilibrium problem is equivalent to a noncooperative game among the populations. Relying on this game formulation, we proposed a multi-population Hessian–Riemannian flow for computing the equilibrium. The flow exploited the geometry of the feasible flow space and preserved the Kirchhoff flow-conservation constraints. We proved convergence of the continuous-time dynamics under the stated assumptions and demonstrated the method on heterogeneous traffic examples through comparisons with benchmark methods, including projected gradient, Gauss–Seidel, and nonlinear programming, as well as through an emissions-related case study.