Solving a Signalized Traffic Intersection Problem with NLP Solvers

Abstract

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 average waiting time over the worst queue, or average waiting time.

A single signalized intersection regulated by pretimed control problem with four traffic streams is considered. This problem is formulated as an MPCC and also as its NLP equivalent reformulation. Some computational experiments using the FilterMPEC, the KNITRO, and the MATLAB optimization toolbox are performed.

This paper is organized as follows. Next section defines the MPCC problem and its NLP reformulation. Some optimal issues are presented in Section 3. The traffic model formulation using two objective functions is described in Section 4. In Section 5, the optimization solvers main characteristics are presented. Numerical experiments using the solvers are reported in Section 6. Some conclusions and future work are carried out in Section 7.

2. Problem Definition

We consider Mathematical Program with Complementarity Constraints (MPCC): min𝑓(𝑥)s.t.𝑐_𝑖(𝑥)𝑐=0,𝑖∈𝐸,_𝑖(𝑥)≥0,𝑖∈𝐼,0≤𝑥₁⟂𝑥₂()≥0,MPCC where 𝑓 and 𝑐 are the nonlinear objective function and the constraint functions, respectively, assumed to be twice continuously differentiable. 𝐸 and 𝐼 are two disjoined finite index sets with cardinalities 𝑝 and 𝑚, respectively. A decomposition 𝑥=(𝑥₀,𝑥₁,𝑥₂) of the variables is used where 𝑥₀∈ℝ^𝑛 (control variables) and (𝑥₁,𝑥₂)∈ℝ^2𝑞 (state variables). The expressions 0≤𝑥₁⟂𝑥₂≥0∶ℝ^2𝑞→ℝ^𝑞 are the 𝑞 complementarity constraints. 𝑄 is a finite index set with cardinality 𝑞.

One attractive way of solving (MPCC) is to replace the complementarity constraints by a set of nonlinear inequalities, such as 𝑥_1𝑗𝑥_2𝑗≤0,  𝑗∈𝑄, and then solve the equivalent nonlinear program (NLP): min𝑓(𝑥)s.t.𝑐_𝑖(𝑥)𝑐=0,𝑖∈𝐸,_𝑖(𝑥)𝑥≥0,𝑖∈𝐼,_1𝑗𝑥_2𝑗𝑥≤0,𝑗∈𝑄,₁≥0,𝑥₂()≥0.MPCC-NLP

3. Optimal Issues

The NLP formulation has no feasible point that strictly satisfies the inequalities. This fact implies that the Mangasarian-Fromovitz constraint qualification (MFCQ) is violated at every feasible point [16]. This failure has consequences: the multiplier set is unbounded, the central path fails to exist, the active constraints normals are linearly dependent, and linearizations of the NLP formulation can be inconsistent arbitrarily close to the solution. Recent developments show that there is a relationship between strong stationarity defined by Scheel and Scholtes and the Karush-Kuhn-Tucker (KKT) points. This relationship established convergence of SQP methods for MPCC formulated as NLP. Some optimality concepts used in this work are based on the study of Fletcher et al. [14].

4. Traffic Model Formulation

In this section, we briefly describe the traffic model and present the MPCC formulation—further details could be consulted in the work of Ribeiro and Simões [17]. The traffic model considers an intersection with four traffic streams: 𝑆₁, 𝑆₂, 𝑆₃, and 𝑆₄ which are controlled by a traffic signal: 𝑇₁, 𝑇₂, 𝑇₃, and 𝑇₄, respectively. A set 𝑆={1,2,3,4} is considered. The intersection presented in Figure 1 is controlled by two phases (A and B). During Phase A, the traffic signals 𝑇₁ and 𝑇₃ have green light and the same occurs in Phase B for 𝑇₂ and 𝑇₄. In both phases, the cycle has 3 stages: green, yellow, and red. The arrival rate of vehicles in traffic stream 𝑆_𝑖 at instant time 𝑡 is 𝜆_𝑖(𝑡) for 𝑖∈𝑆. When the traffic signal 𝑇_𝑖 is green, the departure rate in traffic stream 𝑆_𝑖 at instant time 𝑡 is 𝜇_𝑖(𝑡), and in the case of the traffic being yellow, the departure rate in traffic stream 𝑆_𝑖 at instant 𝑡 is 𝜅_𝑖(𝑡) for 𝑖∈𝑆. Let 𝑡₀, 𝑡₁, …, be the time instants when a change in the traffic signal occurs.

Figure 1 

Intersection.

The duration of the yellow time and clearance time is fixed and set equal to 𝑑_𝑌 and 𝑑_𝐶, respectively. The time instants when traffic signals 𝑇₁ and 𝑇₃ initiate a green period and 𝑇₂ and 𝑇₄ begin a red period are 𝑡₀, 𝑡₂, 𝑡₄, …. The time instants when traffic signals 𝑇₁ and 𝑇₃ initiate a red period and 𝑇₂ and 𝑇₄ begin a green period are 𝑡₁, 𝑡₃, 𝑡₅, …. Thus, 𝑡_2𝑘+1−𝑡_2𝑘=𝑦_𝐺+𝑑_𝑌+𝑑_𝐶 and 𝑡_2𝑘+2−𝑡_2𝑘+1=𝑦_𝑅+𝑑_𝑌+𝑑_𝐶, 𝑘∈ℕ₀.

Therefore, 𝑦_𝐺 represents the green time in traffic signals 𝑇₁ and 𝑇₃ and 𝑦_𝑅 represents the red time at 𝑇₁ and 𝑇₃, and a cycle length is equal to 𝑦_𝐺+𝑦_𝑅+2𝑑_𝑦+2𝑑_𝐶.

The traffic problem has the following MPCC formulation: min⁴_𝑖=11𝑥2𝑁^0𝑖+^𝑁−1_𝑘=11𝑁𝑥_𝑘𝑖+1𝑥2𝑁^𝑁𝑖s.t.0≤𝑥_𝑘≤𝑥_max,𝑦_min≤𝑦_𝑅≤𝑦_max,𝑦_min≤𝑦_𝐺≤𝑦_max,𝑥_2𝑘+1≥𝑥_2𝑘+𝑏₁𝑦_𝐺+𝑏₃⟂𝑥_2𝑘+1≥𝑏₅,𝑥_2𝑘+2≥𝑥_2𝑘+1+𝑏₂𝑦_𝑅+𝑏₄⟂𝑥_2𝑘+2≥𝑏₆,(TP1) where the objective function represents the average queue length over all queues. Since short cycles imply more stops and long cycles causes long delays, maximum and minimum durations for the red and green time (𝑦_𝑅 and 𝑦_𝐺) are considered as simple bound constraints. The number of vehicles in traffic stream 𝑖 at time instant 𝑘 is represented by (𝑥_𝑘)_𝑖. The maximum queue length in each traffic stream is 𝑥_max and 𝑁 is the time periods considered.

Another situation was considered and has the following formulation: minmax_𝑖1𝜆_𝑖1𝑥2𝑁^0𝑖+^𝑁−1_𝑘=11𝑁𝑥_𝑘𝑖+1𝑥2𝑁^𝑁𝑖s.t.0≤𝑥_𝑘≤𝑥_max,𝑦_min≤𝑦_𝑅≤𝑦_max,𝑦_min≤𝑦_𝐺≤𝑦_max,𝑥_2𝑘+1≥𝑥_2𝑘+𝑏₁𝑦_𝐺+𝑏₃⟂𝑥_2𝑘+1≥𝑏₅,𝑥_2𝑘+2≥𝑥_2𝑘+1+𝑏₂𝑦_𝑅+𝑏₄⟂𝑥_2𝑘+2≥𝑏₆.(TP2) In this case, the goal is to minimize the average waiting time over the worst queue, and the constraints are the same as the traffic problem (TP1).

In addition, the following vectors are defined: 𝑥_𝑘=𝐿₁𝑡_𝑘,𝐿₂𝑡_𝑘,𝐿₃𝑡_𝑘,𝐿₄𝑡_𝑘^𝑇,𝑘∈ℕ₀,𝑏₁=𝜆₁−𝜇₁,𝜆₂,𝜆₃−𝜇₃,𝜆₄^𝑇,𝑏₂=𝜆₁,𝜆₂−𝜇₂,𝜆₃,𝜆₄−𝜇₄^𝑇,𝑏₃=𝜆₁−𝜅₁𝑑_𝑌+𝜆₁𝑑_𝐶,𝜆₂𝑑_𝐶+𝑑_𝑌,𝜆₃−𝜅₃𝑑_𝑌+𝜆₃𝑑_𝐶,𝜆₄𝑑_𝐶+𝑑_𝑌^𝑇,𝑏₄=𝜆₁𝑑_𝐶+𝑑_𝑌,𝜆₂−𝜅₂𝑑_𝑌+𝜆₂𝑑_𝐶,𝜆₃𝑑_𝐶+𝑑_𝑌,𝜆₄−𝜅₄𝑑_𝑌+𝜆₄𝑑_𝐶^𝑇,𝑏₅=max𝜆₁−𝜅₁𝑑_𝑌+𝜆₁𝑑_𝐶,𝜆₁𝑑_𝐶,0,max𝜆₃−𝜅₃𝑑_𝑌+𝜆₃𝑑_𝐶,𝜆₃𝑑_𝐶,0^𝑇,𝑏₆=0,max𝜆₂−𝜅₂𝑑_𝑌+𝜆₂𝑑_𝐶,𝜆₂𝑑_𝐶,0,max𝜆₄−𝜅₄𝑑_𝑌+𝜆₄𝑑_𝐶,𝜆₄𝑑_𝐶^𝑇,(1) where for each traffic stream, that is, for 𝑖∈𝑆:(i)𝜆_𝑖 is the average arrival rate; (ii)𝜇_𝑖 is the average departure rate when the traffic signal is green; (iii)𝜅_𝑖 is the average departure rate when the traffic signal is yellow; (iv)𝐿_𝑖(𝑡_𝑘) is the queue length at time instant 𝑘.

These two MPCC problems, (TP1) and (TP2), are reformulated into their NLP equivalent formulation (MPCC-NLP). Next section presents three optimization codes used to solve these two traffic situations in both formulations: (MPCC) and (MPCC-NLP).

5. NLP Solvers

In this section, we present three solvers, two of them are available on the NEOS Server platform [18, 19], and the third one is available on MATLAB software. The NEOS Server is a free internet-based service for solving optimization problems with several solvers representing the state of the art in optimization software.

5.1. FilterMPEC

FilterMPEC arises from the work of Fletcher and Leyffer [20] to solve Mathematical Programs with Equilibrium Constraints (MPECs) in AMPL format [21]. FilterMPEC is an extension of filterSQP, implementing a SQP solver which is suitable for solving medium-scale nonlinearly constrained problems.

5.2. KNITRO

KNITRO was developed for the solution of general nonconvex, nonlinearly constrained optimization problems in AMPL format [22]. It is also effective for problems with complementarity constraints. KNITRO provides three algorithms, two of them use the interior point methods, and the other is based on an active set method.

5.3. MATLAB

The fmincon routine from MATLAB optimization toolbox [23] is a gradient-based method to solve problems with objective and constraint functions twice continuously differentiable. It uses one of four algorithms: active-set, interior point method, SQP, or trust-region-reflective. From these algorithms, we use the SQP algorithm to solve (MPCC-NLP). To connect the modelling language AMPL [24] to the fmnicon routine, a MATLAB mex function was used.

6. Numerical Experiments

This section summarizes the results of the numerical tests, solving the traffic intersection problem with two different approaches: the first approach, used by FilterMPEC and KNITRO, the traffic problem is solved using the (MPCC) formulation; the second approach employed by fmincon routine uses (MPCC-NLP) reformulation. The MATLAB version was 7.11.0 (R2010b), and the computational experiments were made on a 2.26 GHz Intel Core 2 Duo with 8 GB of RAM, MAC OS 10.6.8 operating system. For all the solvers, the default options were used.

Both traffic situations (TP1) and (TP2) were tested using ten instances of 𝜆_𝑖,𝑖∈𝑆, corresponding to problems P1–P10 reported in Table 1. Furthermore, 𝑁=61 was considered, which corresponds to a period of time over thirty cycles. The other parameters used are 𝜇_𝑖=1800veh/h,𝜅_𝑖𝑥=720veh/h,𝑖∈𝑆,_0𝑘=2%𝜆_𝑘𝑥_0𝑗=1%𝜆_𝑘𝑥for𝑘=1,3,𝑗=2,4,_maxi𝑑=25𝑖∈𝑆,_𝑌=3𝑠,𝑑_𝐶𝑦=2𝑠,_min=7𝑠,𝑦_max=60𝑠.(2)

Table 1 

10 instances.

Tables 2 and 3 report the optimization results of traffic problem (TP1) and traffic problem (TP2), respectively: the red split time (𝑦_𝑅), the green split time (𝑦_𝐺), and objective function value (obj) for the three solvers.

Table 2 

(TP1) results.

Table 3 

(TP2) results.

Table 4 presents the number of iterations (it) achieved by the solvers for both traffic situations (TP1) and (TP2). Among the analysed solvers, FilterMPEC is the one that presents the best performance solving the traffic situation (TP1). For traffic situation (TP2), KNITRO needs more iterations than the others solvers.

Table 4 

Number of iterations.

7. Conclusions and Future Work

A traffic model was studied and codified in AMPL language which could be easily connected to nonlinear programming solvers. Two approaches were used to solve the traffic problem: the (MPCC) formulation, solved by FilterMPEC and KNITRO on NEOS Server and the reformulation (MPCC-NLP) solved by fmincon routine from MATLAB. The model for traffic situations (TP1) and (TP2) was efficiently solved by these solvers using both approaches (MPCC) and (MPCC-NLP).

As a first stage of future work, we intend to introduce a probabilistic distribution in arrival rate instead of considering it as a constant value. Another idea is to allow that vehicles to turn (right or left) and not follow just ahead. With minor changes, other objective functions can be easily tested: average queue length over the worst queue, worst case queue length and average waiting time over all queues. Another goal of this work is to collect real data from a signalized traffic intersection in a city in order to test the model in a real situation.

Acknowledgments

This work is funded by FEDER funds through Operational Programme for Competitiveness Factors—COMPETE and National Funds through FCT—Foundation for Science and Technology in Project scope: FCOMP-01-0124-FEDER-022674 (R&D unit Algoritmi). The work is also supported by PEst-OE/MAT/UI4080/2011. The authors are very grateful to all the financial support.