Optimization of continuous-flow over traffic networks with fundamental diagram constraints

TL;DR

This work addresses congestion-aware transport on networks by embedding a fundamental diagram into a dynamic optimal transport framework on graphs. Edge momentum is treated as a controllable variable with capacity bounds $0 \le m \le \mathcal{Q}_{e}(\bar{\rho}_{e})$, and a midpoint time discretization yields a convex program with a unique minimizer, solvable via an augmented-Lagrangian/ADMM scheme. Existence and uniqueness proofs support the theoretical foundation, while the solver exploits sparsity to scale to city-scale networks. Numerical experiments on a single-lane line and a city-planar Athens graph demonstrate realistic congestion patterns, capacity adherence, and convergence to convex baselines, underscoring practical potential for planning and closed-loop control in large-scale transportation systems.

Abstract

Optimal transport (OT) theory provides a principled framework for modeling mass movement in applications such as mobility, logistics, and economics. Classical formulations, however, generally ignore capacity limits that are intrinsic in applications, in particular in traffic flow problems. We address this limitation by incorporating fundamental diagrams into a dynamic continuous-flow OT model on graphs, thereby including empirical relations between local density and maximal flux. We adopt an Eulerian kinetic action on graphs that preserves displacement interpolation in direct analogy with the continuous theory. Momentum lives on edges and density on nodes, mirroring road-network semantics in which segments carry speed and intersections store mass. The resulting fundamental-diagram-constrained OT problem preserves mass conservation and admits a convex variational discretization, yielding optimal congestion-aware traffic flow over road networks. We establish the existence and uniqueness of the optimal flow with sources and sinks, and develop an efficient convex optimization method. Numerical studies begin with a single-lane line network and scale to a city-level road network.

Figures (5)

• Figure 1: Greenshields fundamental diagram. The blue line shows the velocity-density relation, while the red curve shows the momentum (flux) as a function of density. The optimal operating point $(\rho_{\text{opt}}, m_{\text{opt}})$ corresponds to the maximum flux, attained at half the jam density $\hat{\rho}/2$ with velocity $v_{\text{opt}}$.
• Figure 2: Evolution of densities on a 30-node line graph with $8$ snapshots at evenly spaced times $t=0/7,1/7,\dots,7/7$. Nodes are colored from blue (low density) to red (high density), with darker shades indicating higher values. Edges are drawn as semi-transparent blue bars, with thickness proportional to total momentum flow across each link, scaled consistently across snapshots. The dynamics are obtained from the midpoint discretization of the convex formulation, with symmetry enforced by averaging forward and backward momentum in time. $\hat{\rho} = 0.15$ and $v_0=3$.
• Figure 3: Top row: congestion-aware transport on a directed network with $n=291$ nodes and $m=614$ directed edges. Bottom row: the same setting without the fundamental-diagram constraint. Snapshots are shown at $t\in\{0,\tfrac{1}{7},\tfrac{2}{7},\tfrac{3}{7},\tfrac{4}{7},\tfrac{5}{7},\tfrac{6}{7},1\}$ using a midpoint discretization with k=7 steps. For the top row, Greenshields capacities $\mathcal{Q}_e(\rho)=v_0\,\rho\!\left(1-\rho/\hat{\rho}\right)$ with $\hat{\rho}=0.05$ and $v_0=2$ are enforced on interior steps, and the endpoint frames are unconstrained. The bottom row uses the same solver and time grid but omits the capacity constraint. Node color and size encode density, with blue near zero and red at higher values and marker area proportional to mass.
• Figure 4: ADMM convergence on a line with six snapshots (FD inactive). The curve shows the kinetic objective versus iteration. The dashed line marks the CVX optimum.
• Figure 5: ADMM convergence with Greenshields FD active ($v_0=1$, $\hat{\rho}=0.10$). The projection step enforces the FD constraint at every iteration; the objective approaches the CVX benchmark under the same ADMM parameters as in Figure \ref{['fig:convergence_admm']}.

Theorems & Definitions (6)

• Definition 1: Fundamental diagram
• Definition 2: LWR model
• Definition 3: Edge-wise Fundamental Diagram
• Proposition 1: Uniqueness
• proof
• Remark 1: Time varying fundamental diagram