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

Abstract

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.

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Acknowledgments

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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Correspondence to Jack Reilly.

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Communicated by Emilio Frazzoli.

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Reilly, J., Samaranayake, S., Delle Monache, M.L. et al. Adjoint-Based Optimization on a Network of Discretized Scalar Conservation Laws with Applications to Coordinated Ramp Metering. J Optim Theory Appl 167, 733–760 (2015). https://doi.org/10.1007/s10957-015-0749-1

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Keywords

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

Mathematics Subject Classification

  • 35L65
  • 90-08