Journal of Optimization Theory and Applications

, Volume 167, Issue 2, pp 733–760 | Cite as

Adjoint-Based Optimization on a Network of Discretized Scalar Conservation Laws with Applications to Coordinated Ramp Metering

  • Jack Reilly
  • Samitha Samaranayake
  • Maria Laura Delle Monache
  • Walid Krichene
  • Paola Goatin
  • Alexandre M. Bayen


The adjoint method provides a computationally efficient means of calculating the gradient for applications in constrained optimization. In this article, we consider a network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function. After discretizing the corresponding partial differential equation models via the Godunov scheme, we detail the computation of the gradient of the discretized system with respect to the control parameters and show that the complexity of its computation scales linearly with the number of discrete state variables for networks of small vertex degree. The method is applied to the problem of coordinated ramp metering on freeway networks. Numerical simulations on the I15 freeway in California demonstrate an improvement in performance and running time compared with existing methods. In the context of model predictive control, the algorithm is shown to be robust to noise in the initial data and boundary conditions.


Control of discretized PDEs Network of hyperbolic conservation laws Adjoint-based optimization Transportation engineering Ramp metering 

Mathematics Subject Classification

35L65 90-08 



The authors have been supported by the California Department of Transportation under the Connected Corridors program, CAREER Grant CNS-0845076 under the project ‘Lagrangian Sensing in Large Scale Cyber-Physical Infrastructure Systems’, the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013)/ERC Grant Agreement No. 257661, the INRIA associated team ‘Optimal REroute Strategies for Traffic managEment’ and the France-Berkeley Fund under the project ‘Optimal Traffic Flow Management with GPS Enabled Smartphones’.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.Inria Sophia Antipolis - MéditerranéeSophia AntipolisFrance

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