### Optimization for a special class of traffic flow models: Combinatorial and continuous approaches

Simone Göttlich, Oliver Kolb, Sebastian Kühn
2014 Networks and Heterogeneous Media
In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions. 2010 Mathematics Subject Classification. 90B20, 49K20, 90C11. 315 316 SIMONE GÖTTLICH,
more » ... LIVER KOLB AND SEBASTIAN KÜHN leads to numerically solving the finite-dimensional optimality system, i.e. the socalled discretize-then-optimize approach, or the interpretation as a mixed-integer programming model (MIP). However, it remains the question of detecting local or global optima. We know from the theory of linear programming, in particular the strong duality theorem [37] , that under certain circumstances a global optimum can be reached. This is usually not the case for the adjoint calculus. The solution of the optimality system via gradient methods often gets stuck in local optima. In this work we intend to close the gap between the two solution procedures that first appear to be different. To do so, we start in section 2 with the traffic flow network model where the evolution of traffic density on roads is governed by the linearized Lighthill-Whitham-Richards (LWR) equations, see [9] . For the coupling conditions at the intersections, we stick to the ones presented by Coclite-Garavello-Piccoli [5] . In a next step, we discretize the full network model in space and time and formally derive the adjoint equations and the mixed-integer model (MIP) as well, see sections 3-5. In section 6, we point out the equivalence of both approaches by comparing the dual variables of the MIP and the adjoint variables for a linear network formally and numerically. Furthermore, a more complex network is treated numerically. 2. Traffic flow network model. In this section, we briefly review the main modeling issues, see [5, 23, 26] .