A convex reformulation solution approach for the joint perimeter control and route guidance problem

Macroscopic traffic management schemes, such as Perimeter Control and Route Guidance, have attracted a lot of attention as primary traffic control strategies for congestion alleviation. Macroscopic methods take advantage of the Macroscopic Fundamental Diagram to capture traffic dynamics within an urban area. Such schemes often result in non-linear and nonconvex mathematical programs that are solved with standard non-linear optimization solvers. Nonetheless, non-linear solvers can yield low-quality solutions, are slow and unreliable, and provide no information on the quality of the derived solution. Building upon earlier macroscopic schemes, the contribution of this work is the development of a novel solution methodology for route guidance with perimeter control. The proposed methodology constructs convex outer approximations of all nonlinear constraint sets of the problem to derive: (i) a tight lower bound formulation and (ii) an iterative convexification procedure that provides feasible and upper-bound solutions. The resulting lower and upper bound formulations are solved using Linear Programming producing fast, high quality, and reliable solutions while also providing guaranteed optimality gaps. Macroscopic simulations demonstrate that the proposed methodology executes 2 to 5 times faster than a state-of-the-art non-linear solver and offers an optimality gap of less than 3.9 % in all considered cases.