A new perspective on dynamic network flow problems via port-Hamiltonian systems

TL;DR

This paper introduces a novel approach to dynamic network flow problems by modeling them as port-Hamiltonian systems, enabling the use of optimal control techniques and gradient algorithms for efficient solutions.

Contribution

It formulates dynamic minimum cost flow problems within the port-Hamiltonian framework and develops an adjoint-based gradient descent algorithm for solving them.

Findings

Proved well-posedness of port-Hamiltonian system models.

Derived an optimality system for control characterization.

Validated the approach with numerical experiments on static and dynamic flows.

Abstract

We suggest a global perspective on dynamic network flow problems that takes advantage of the similarities to port-Hamiltonian dynamics. Dynamic minimum cost flow problems are formulated as open-loop optimal control problems for general port-Hamiltonian systems with possibly state-dependent system matrices. We prove well-posedness of these systems and characterize optimal controls by the first-order optimality system, which is the starting point for the derivation of an adjoint-based gradient descent algorithm. Our theoretical analysis is complemented by a proof of concept, where we apply the proposed algorithm to static minimum cost flow problems and dynamic minimum cost flow problems on a simple directed acyclic graph. We present numerical results to validate the approach.