Solving a Signalized Traffic Intersection Problem with NLP Solvers

Mathematical Programs with Complementarity Constraints (MPCC) finds many applications in areas such as engineering design, economic equilibrium, and mathematical theory itself. In this work, we consider a queuing system model resulting from a single signalized traffic intersection regulated by pretimed control in an urban traffic network. The model is formulated as an MPCC problem and may be used to ascertain the optimal cycle and the green split allocation. This MPCC problem is also formulated as its NLP equivalent reformulation. The goal of this work is to solve the problem, using both MPCC and NLP formulations, minimizing two objective functions: the average queue length over all queues and the average waiting time over the worst queue. The problem was codified in AMPL and solved using some optimization software packages. 1. Introduction Mathematical Programs with Complementarity Constraints (MPCC) is a subclass of more general Mathematical Programs with Equilibrium Constraints (MPEC). These kind of constraints may come as a game, a variational inequality, or as stationary conditions of an optimization problem. The main applications areas are engineering and economics [1–3]. They are so widespread in these areas because the concept of complementarity is synonymous with the notion of system equilibrium. They are very difficult to solve as the usual constraint qualifications necessary to guarantee the algorithms convergence fail in all feasible points [4]. This complexity is caused by the disjunctive nature of the complementarity constraints. Some nonlinear approaches to solve MPCC have been proposed, starting with the smoothing scheme [5, 6], the regularization scheme [7, 8], the interior point methods [9], the penalty approaches [10–12], and the “elastic mode” for nonlinear programming in conjunction with a sequential quadratic programming (SQP) algorithm [13]. In Fletcher et al.’s study [14], the quadratic convergence of SQP is guaranteed, near a stationary point, under relatively mild conditions. As the number of vehicles and the need for transportation grow, traffic light control can be used to control the flow of the traffic in urban environments. De Schutter and De Moor [15] study the optimal traffic control problem of a two two-way streets intersection. These authors derive an approximate model that describes the evolution of the queues lengths as a continuous function of time. Starting from this model it is possible to compute the traffic light switching scheme that minimizes a criterion such as average queue length over all queues, the