Analytical Reduction for Traffic Networks
Abstract
Repeated traffic assignment is a major computational bottleneck in bilevel transport optimization and backcasting applications. Existing approaches address this difficulty through equilibrium sensitivity analysis, network contraction, or calibration of link travel-time functions, but these approaches generally either construct reduced representations from equilibrium sensitivities or retain the original network while repeatedly solving the lower-level assignment. This paper considers a complementary problem: given a strongly reduced representation of a large reference network, how can its link parameters be calibrated so that it reproduces the equilibrium origin-destination (OD) travel times of the reference network at low computational cost. To address this problem, we build on the Linear Analytical Model (LAM) introduced in a previous paper. The LAM linearises the link cost functions around a reference congestion state and retains a fixed candidate-path support, allowing the equilibrium conditions to be expressed through a reduced linear system and providing a closed-form relationship between link parameters, OD demand, OD travel times, and link flows. We exploit this structure to formulate the calibration of the reduced network as a constrained nonlinear least-squares problem. The decision variables are the linearised free-flow travel times and capacities of the reduced links, while the target values are the equilibrium OD travel times obtained on the full network. The main contribution is the analytical differentiation of this calibration problem. We derive the exact gradient of the OD travel-time matching objective and the exact Jacobian of the flow constraints by differentiating the Schur-complement structure underlying the LAM. Unlike finite-difference or simulation-based approaches, the proposed formulation does not require repeated traffic assignments or repeated evaluations of the full equilibrium model to obtain parameter sensitivities. The resulting derivatives enable the use of gradient-based constrained optimization for the reduced-network calibration problem. The analytical derivatives are first validated against high-order finite differences, with a maximum relative error below 10 -8 . The methodology is then evaluated on the Sioux Falls benchmark and on a strongly reduced road network of Lyon, France, for four combinations of OD resolution and demand level. The calibrated reduced networks reproduce the target OD travel times with R2 values between 0.92 and 0.98 in the Lyon experiments, while each optimization iteration requires only a few seconds on networks containing several hundred links. The results demonstrate that analytical differentiation of the LAM can turn reducednetwork calibration into a computationally tractable gradient-based problem, providing a practical framework for repeatedly embedding reduced transport models in large-scale bilevel and backcasting applications.
Related
- Linear Analytical Model for Static Traffic Assignment: Explicit Formulation of OD Travel Times and Link Flows
- Data-driven calibration of a two-dimensional traffic flow model: Dimension reduction and optimization strategies
- MIXSim: A multiclass first order traffic model with composition-dependent dynamics for mixed-autonomy urban networks
- State-Dependent Phase-Type Queueing Networks for Urban Traffic Signal Control
- Stochastic Gradient Descent for Traffic and Pedestrian Models via Neural Networks and Parameter Identification
BibTeX
@misc{leroux2026analytical,
title = {{Analytical Reduction for Traffic Networks}},
author = {Baptiste Leroux and Antonio Sciarretta and Miloš Balać},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05851388v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytical Reduction for Traffic Networks
AU - Baptiste Leroux
AU - Antonio Sciarretta
AU - Miloš Balać
PY - 2026
DA - 2026/11/04
PB - HAL
UR - https://hal.science/hal-05851388v1
AB - Repeated traffic assignment is a major computational bottleneck in bilevel transport optimization and backcasting applications. Existing approaches address this difficulty through equilibrium sensitivity analysis, network contraction, or calibration of link travel-time functions, but these approaches generally either construct reduced representations from equilibrium sensitivities or retain the original network while repeatedly solving the lower-level assignment. This paper considers a complementary problem: given a strongly reduced representation of a large reference network, how can its link parameters be calibrated so that it reproduces the equilibrium origin-destination (OD) travel times of the reference network at low computational cost. To address this problem, we build on the Linear Analytical Model (LAM) introduced in a previous paper. The LAM linearises the link cost functions around a reference congestion state and retains a fixed candidate-path support, allowing the equilibrium conditions to be expressed through a reduced linear system and providing a closed-form relationship between link parameters, OD demand, OD travel times, and link flows. We exploit this structure to formulate the calibration of the reduced network as a constrained nonlinear least-squares problem. The decision variables are the linearised free-flow travel times and capacities of the reduced links, while the target values are the equilibrium OD travel times obtained on the full network. The main contribution is the analytical differentiation of this calibration problem. We derive the exact gradient of the OD travel-time matching objective and the exact Jacobian of the flow constraints by differentiating the Schur-complement structure underlying the LAM. Unlike finite-difference or simulation-based approaches, the proposed formulation does not require repeated traffic assignments or repeated evaluations of the full equilibrium model to obtain parameter sensitivities. The resulting derivatives enable the use of gradient-based constrained optimization for the reduced-network calibration problem. The analytical derivatives are first validated against high-order finite differences, with a maximum relative error below 10 -8 . The methodology is then evaluated on the Sioux Falls benchmark and on a strongly reduced road network of Lyon, France, for four combinations of OD resolution and demand level. The calibrated reduced networks reproduce the target OD travel times with R2 values between 0.92 and 0.98 in the Lyon experiments, while each optimization iteration requires only a few seconds on networks containing several hundred links. The results demonstrate that analytical differentiation of the LAM can turn reducednetwork calibration into a computationally tractable gradient-based problem, providing a practical framework for repeatedly embedding reduced transport models in large-scale bilevel and backcasting applications.
ER -