Traffic Equilibrium Problem Research Articles

On strong Lagrange duality for weighted traffic equilibrium problem

The weighted traffic equilibrium problem introduced in [17], in which the equilibrium conditions have been expressed in terms of a weighted variational inequality, studies a transportation network in presence of congestion. For such a problem, existence and regularity theorems have been proved in [8]. In this paper, we analyze the dual problem and characterize the weighted traffic equilibrium solutions by means of Lagrange multipliers, which allow to describe the behavior of the weighted transportation network.

A semismooth Newton method for traffic equilibrium problem with a general nonadditive route cost

Computing traffic equilibria with a general nonadditive route cost disutility function is considered in this paper. Following the user equilibrium (UE) condition, that is, no driver can unilaterally change route to achieve less travel costs, the traffic equilibrium problem (TEP) can be formulated as a nonlinear complementary problem (NCP). In this paper, we propose a semismooth Newton method with a penalized Fischer–Burmeister (PFB) NCP function to solve the NCP formulation of the TEP, and also, we investigate the properties of the proposed method. Numerical results are provided and compared with the classical TEP with additive route cost functions. The results show the algorithm can achieved substantially better performance than the existing approaches. A sensitivity analysis is also conducted to examine the parameter of the proposed nonadditive route cost function.

Approximation Techniques for Transportation Network Design Problem under Demand Uncertainty

Conventional transportation network design problems treat origin-destination (OD) demand as fixed, which may not be true in reality. Some recent studies model fluctuations in OD demand by considering the first and the second moment of the system travel time, resulting in stochastic and robust network design models, respectively. Both of these models need to solve the traffic equilibrium problem for a large number of demand samples and are therefore computationally intensive. In this paper, three efficient solution-approximation approaches are identified for addressing demand uncertainty by solving for a small sample size, reducing the computational effort without much compromise on the solution quality. The application and the performance of these alternative approaches are reported. The results from this study will help in deciding suitable approximation techniques for network design under demand uncertainty.

An LQP-based descent method for structured monotone variational inequalities

This paper proposes a descent method to solve a class of structured monotone variational inequalities. The descent directions are constructed from the iterates generated by a prediction-correction method [B.S. He, Y. Xu, X.M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Comput. Optim. Appl. 35 (2006) 19–46], which is based on the logarithmic-quadratic proximal method. In addition, the optimal step-sizes along these descent directions are identified to accelerate the convergence of the new method. Finally, some numerical results for solving traffic equilibrium problems are reported.

Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs

In this paper, we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. The path cost function considered is comprised of two attributes, travel time and toll, that are combined into a nonlinear generalized cost. Travel demand is determined endogenously according to a travel disutility function. Travelers choose routes with the minimum overall generalized costs. The algorithm involves two components: a bicriteria shortest path routine to implicitly generate the set of non-dominated paths and a projection and contraction method to solve the nonlinear complementarity problem (NCP) describing the traffic equilibrium problem. Numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems.

Approximate analytical expressions for transportation network performance under demand uncertainty

This paper deals with accounting for demand uncertainty in the symmetric traffic equilibrium problem. We initially show that by not accounting for long term demand uncertainty, the total network travel time obtained from the traffic equilibrium problem is substantially underestimated. Then, we develop closed form approximate analytical expressions for expected value and the variance of the network performance (a measure of total system travel time) resulting from uncertain demand. We empirically show the relationship between the uncertain OD demand and the link/path flows in a network with the assumption that the long term demand is normally distributed. Illustrative numerical experiments are conducted on multiple test networks and the total network impact in terms of total system travel time is compared with the derived analytical expressions. It is observed that the expressions for the expected value match very closely to the true system impact for small networks and becomes an approximation as the network size grows. The robustness (variance) expression is observed to be more deviant for larger networks than the expected value. However, the ordinal ranking of robustness was found to be consistent with the true network robustness. Further, recent work on OD correlations, applicability on large networks and directions for future research are discussed. The analytical expressions should allow the characterization of network uncertainty when link flow uncertainty is known.

Continuity Results for a Class of Variational Inequalities with Applications to Time-Dependent Network Problems

We consider a class of monotone variational inequalities in which both the operator and the convex set are parametrized by continuous functions. Under suitable assumptions, we prove the continuity of the solution with respect to the parameter. As an important application, we consider the case of finite dimensional variational inequalities on suitable polyhedra. We demonstrate the applicability of our results to time dependent traffic equilibrium problem.

An improved LQP-based method for solving nonlinear complementarity problems

The well-known logarithmic-quadratic proximal (LQP)method has motivated a number of efficient numerical algorithms for solving nonlinear complementarity problems (NCPs). In this paper,we aim at improving one of them, i.e., the LQP-based interior prediction-correction method proposed in [He, Liao and Yuan, J. Comp. Math., 2006, 24(1): 33–44], via identifying more appropriate step-sizes in the correction steps. Preliminary numerical results for solving some NCPs arising in traffic equilibrium problems are reported to verify the theoretical assertions.

Lipschitz Continuity Results for a Class of Variational Inequalities and Applications: A Geometric Approach

We consider a class of parametric variational inequalities which, under suitable assumptions, admit an equivalent integral formulation. We study the Lipschitz continuity of the solution and treat in detail the case in which the parametric constraint set is a polytope. Finally, the results obtained are applied to the time-dependent traffic equilibrium problem.

Self-adaptive projection-based prediction-correction method for constrained variational inequalities

The problems studied in this paper are a class of monotone constrained variational inequalities VI (S, f) in which S is a convex set with some linear constraints. By introducing Lagrangian multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. We focus on the mapping f that does not have an explicit form. Therefore, only its function values can be employed in the numerical methods. The number of iterations is significantly dependent on a parameter that balances the primal and dual variables. To overcome potential difficulties, we present a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Convergence of the proposed method is proved under mild conditions. Preliminary numerical experiments including some traffic equilibrium problems indicate the effectiveness of the proposed methods.

Continuous optimization and combinatorial optimization

Optimization is a classical yet fast-growing field. It has been playing vital roles in many areas of Science, Engineering, Business, and our daily life. In recent years, the vast applications of optimization and the tremendous advancement in computing speed (even parallel computers) have created many new research frontiers and challenges both theoretically and numerically for the researchers and practitioners in the field of optimization. In this special issue, papers in continuous optimization as well as combinatorial optimization cover theoretical investigations, numerical experiments, and practical applications. The paper by Xiaoling Fu and Bingsheng He studies a class of monotone variational inequalities with linear constraints. Strong theoretical achievement and some efficient numerical scheme have been obtained. Numerical experiments, including some traffic equilibrium problems, are presented. In the paper by Min Li and Xiao-Ming Yuan, they focus on some nonlinear complementarity problems; by using the logarithmic-quadratic proximal method, a new efficient computing scheme is proposed. Solid convergence and attractive numerical results are presented. The paper by Hongyu Zhang and Yiju Wang proposes a new type of solution methods for the split feasibility problem. Its global convergence is proved under mild conditions, and preliminary numerical results are reported to show the efficiency of the proposed method. The next three papers focus on the combinatorial optimization. The paper by Qizhi Fang, Rudolf Fleischer, Jian Li, and Xiaoxun Sun is devoted to core stability

The study of traffic equilibrium problems with capacity constraints of arcs

In this paper, we study traffic equilibrium problems with capacity constraints of arcs and derive a sufficient condition of weak vector equilibrium flows. Based on this sufficient condition, we establish the existence and essential components of the solution set for traffic equilibrium problems with capacity constraints of arcs.

Road pricing for congestion control with unknown demand and cost functions

It is widely recognized that precise estimation of road tolls for various pricing schemes requires a few pieces of information such as origin–destination demand functions, link travel time functions and users’ valuations of travel time savings, which are, however, not all readily available in practice. To circumvent this difficulty, we develop a convergent trial-and-error implementation method for a particular pricing scheme for effective congestion control when both the link travel time functions and demand functions are unknown. The congestion control problem of interest is also known as the traffic restraint and road pricing problem, which aims at finding a set of effective link toll patterns to reduce link flows to below a desirable target level. For the generalized traffic equilibrium problem formulated as variational inequalities, we propose an iterative two-stage approach with a self-adaptive step size to update the link toll pattern based on the observed link flows and given flow restraint levels. Link travel time and demand functions and users’ value of time are not needed. The convergence of the iterative toll adjustment algorithm is established theoretically and demonstrated on a set of numerical examples.

Weighted variational inequalities in non-pivot Hilbert spaces with applications

We introduce variational inequalities defined in non-pivot Hilbert spaces and we show some existence results. Then, we prove regularity results for weighted variational inequalities in non-pivot Hilbert space. These results have been applied to the weighted traffic equilibrium problem. The continuity of the traffic equilibrium solution allows us to present a numerical method to solve the weighted variational inequality that expresses the problem. In particular, we extend the Solodov-Svaiter algorithm to the variational inequalities defined in finite-dimensional non-pivot Hilbert spaces. Then, by means of a interpolation, we construct the solution of the weighted variational inequality defined in a infinite-dimensional space. Moreover, we present a convergence analysis of the method.

On the regularity of retarded equilibria in time-dependent traffic equilibrium problems

The paper is divided into two parts. In the first part, the retarded link formulation traffic equilibrium problem is presented and a variational inequality formulation is provided. In the second part, we consider a class of strictly monotone evolutionary variational inequalities and present a regularity result for this class. Then, we apply the result to equilibrium problems with data depending explicitly on time. The regularity results allow us to provide a discretization procedure for the calculation of the approximated solution to the retarded time-dependent traffic network equilibrium problem.

Sensitivity of the Traffic Flows in Transportation Networks

Obtaining the equilibrium flows in transportation networks based on optimizing users’ cost is one of the important problems in urban transportation planning. Therefore, finding accurate equilibrium flows in a short time is a significant issue. The required time to solve the traffic equilibrium problem and the accuracy of the obtained results depend on the information level about the transportation system. Using more detail increases the cost of data collection and, dimensions of the problem and, therefore, creates difficulties in solving it. Certainly, a detail level of data should be proportional to the accuracy level that is required. Collecting additional data does not necessarily improve the accuracy level of the results and may only increase the design costs. In this paper, the sensitivity analysis of the equilibrium flows in traffic assignment models with respect to changes in some important transportation network attributes is considered. These attributes include: the accuracy level in measuring link specifications (e.g., width), diversity of travel time functions, intersection delay functions, the method of connecting the zone centroids to a transportation network and number of traffic zones.

Weighted traffic equilibrium problem in non pivot Hilbert spaces

We introduce a weighted traffic equilibrium problem in a non-pivot Hilbert space and prove the equivalence between a weighted Wardrop condition and a variational inequality. As an application we show, using some recent results of the Senseable Laboratory at MIT, how wireless devices can be used to optimize the traffic equilibrium problem.

Stochastic Nonlinear Complementarity Problem and Applications to Traffic Equilibrium under Uncertainty

The expected residual minimization (ERM) formulation for the stochastic nonlinear complementarity problem (SNCP) is studied in this paper. We show that the involved function is a stochastic R0 function if and only if the objective function in the ERM formulation is coercive under a mild assumption. Moreover, we model the traffic equilibrium problem (TEP) under uncertainty as SNCP and show that the objective function in the ERM formulation is a stochastic R0 function. Numerical experiments show that the ERM-SNCP model for TEP under uncertainty has various desirable properties.

A projection-based prediction–correction method for structured monotone variational inequalities

In this paper, we propose a modified prediction–correction method for structured monotone variational inequalities. This method is based on a reformulation of the normal prediction–correction method. Each iteration of the new method contains two predictions and a correction. At each iteration a new strategy is applied to obtain a proper step length. All the computing processes are easily implemented and the global convergence is also presented under mild assumptions. Numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice.

The multi-class, multi-criterion traffic equilibrium and the efficiency of congestion pricing

We consider the effect of the so-called second-best tolls on the price of anarchy of the traffic equilibrium problem where there are multiple classes of users with a discrete set of values of time. We derive several bounds of the price of anarchy for this problem when the tolls are considered and not considered as part of the system cost, with the time-based criterion and the cost-based criterion, respectively. All the bounds give us useful information on the effect of the tolls, which can be used to design network toll schemes.