Traffic Network Equilibrium Problem Research Articles

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

With the progress of society and the diversification of transportation modes, people are faced with more and more complicated travel choices, and thus, multi-criteria route choosing optimization problems have drawn increased attention in recent years. A number of multi-criteria traffic network equilibrium problems have been proposed, but most of them do not involve data uncertainty nor computational methods. This paper focuses on the methods for solving robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of the robust vector equilibrium and the robust vector equilibrium with respect to the worst case are introduced, respectively. For the robust vector equilibrium, an equivalent min–max optimization problem is constructed. A direct search algorithm, in which the step size without derivatives and redundant parameters, is proposed for solving this min–max problem. In addition, we construct a smoothing optimization problem based on a variant version of ReLU activation function to compute the robust weak vector equilibrium flows with respect to the worst case and then find robust vector equilibrium flows with respect to the worst case by using the heaviside step function. Finally, extensive numerical examples are given to illustrate the excellence of our algorithms compared with existing algorithms. It is shown that the proposed min–max algorithm may take less time to find the robust vector equilibrium flows and the smoothing method can more effectively generate a subset of the robust vector equilibrium with respect to the worst case.

Convergence study on the logarithmic-quadratic proximal regularization of strictly contractive Peaceman–Rachford splitting method with larger step-size

Recently, a strictly contractive Peaceman–Rachford splitting method with logarithmic-quadratic proximal regularization (SPRSM-LQP) was proposed for solving two-block separable convex minimization model. In practical applications, however, the smaller step-size should be strongly avoided. So we actually have the desire of seeking larger step-size whenever possible in order to accelerate the numerical performance. In this paper, we combine Fortin and Glowinski's accelerating techniques with the SPRSM-LQP. Thus a new algorithm with larger step-size is proposed. Under the same assumptions as the SPRSM-LQP, we establish the global convergence of its larger step-size counterpart. Moreover, preliminary numerical results show that the proposed method on a traffic network equilibrium problem is reliable and more efficient with larger step-size.

On the stability of approximate solutions to set-valued equilibrium problems

In this paper, we consider set-valued equilibrium problems in Hausdorff locally convex topological vector spaces. Based on linear scalarization techniques for sets, we study sufficient conditions for the stability of approximate solutions to such problems. Variational inequalities with equilibrium constraints and weak traffic network equilibrium problems are also discussed as applications of the main results.

Traffic network equilibrium problems with demands uncertainty and capacity constraints of arcs by scalarization approaches

This paper focuses on the vector traffic network equilibrium problem with demands uncertainty and capacityconstraints of arcs, in which, the demands are not exactly known and assumed as a discrete set that contains finite scenarios.For different scenario, the demand may be changed, which seems much more reasonable in practical programming.By using the linear scalarization method, we introduce several definitions of parametric equilibriumflows and reveal their mutual relations. Meanwhile, the relationships between the scalar variational inequality as well as the (weak) vector equilibrium flows are explored, meanwhile, some necessary and sufficient conditions that ensure the (weak) vector equilibrium flows are also considered.Additionally, by means of nonlinear scalarization functionals, twokinds of equilibrium principles are derived. All of the derived conclusions contain the demands uncertainty and capacity constraints of arcs, thus the resultsproposed in this paper improved some existing works. Finally, some numericalexamples are employed to show the merits of the improved conclusions.

Traffic equilibrium with a continuously distributed bound on travel weights: the rise of range anxiety and mental account

A new traffic network equilibrium problem with continuously distributed bounds on path weights is introduced in this paper, as an emerging modeling tool for evaluating traffic networks in which the route choice behavior of individual motorists is subject to some physical or psychological upper limit of a travel weight. Such a problem may arise from at least two traffic network instances. First, in a traffic network serving electric vehicles, the driving range of these vehicles is subject to a distance constraint formed by onboard battery capacities and electricity consumption rates as well as network-wide battery-recharging opportunities, which cause the range anxiety issue in the driving population. Second, in a tolled traffic network, while drivers take into account both travel time and road toll in their route choice decisions, many of them implicitly or explicitly set a budget constraint in their mental account for toll expense, subject to their own income levels and other personal and household socio-economic factors. In both cases, we model the upper limit of the path travel weight (i.e., distance or toll) as a continuously distributed stochastic parameter across the driving population, to reflect the diverse heterogeneity of vehicle- and/or motorist-related travel characteristics. For characterizing this weight-constrained network equilibrium problem, we proposed a convex programming model with a finite number of constraints, on the basis of a newly introduced path flow variable named interval path flow rate. We also analyzed the problem’s optimality conditions for the case of path distance limits, and studied the existence of optimal tolls for the case of path toll limits. A linear approximation algorithm was further developed for this complex network equilibrium problem, which encapsulates an efficient weight-constrained k-minimum time path search procedure to perform the network loading. Numerical results obtained from conducting quantitative analyses on example networks clearly illustrate the applicability of the modeling and solution methods for the proposed problem and reveal the mechanism of stochastic weight limits reshaping the network equilibrium.

Painlevé–Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problems

This paper is concerned with the stability of semi-infinite vector optimization problems (SIVOP) under functional perturbations of both objective functions and constraint sets. First, we establish the Berge-lower semicontinuity and Painleve–Kuratowski convergence of the constraint set mapping. Then, using the obtained results, we obtain sufficient conditions of Painleve–Kuratowski stability for approximate efficient solution mapping and approximate weakly efficient solution mapping to the (SIVOP). Furthermore, an application to the traffic network equilibrium problems is also given.

An empirical interpolation approach to reduced basis approximations for variational inequalities

ABSTRACTVariational inequalities (VIs) are pervasive in mathematical modelling of equilibrium and optimization problems in engineering and science. Examples of applications include traffic network equilibrium problems, financial equilibrium, obstacle problems, lubrication phenomena and many others. Since these problems are computationally expensive to solve, we focus here on the development of model order reduction techniques, in particular the reduced basis technique. Reduced basis techniques for the approximation of solutions to elliptic VIs have been developed in the last few years. These methods apply to VIs of the so-called first kind, i.e. problems that can be equivalently described by a minimization of a functional over a convex set. However, these recent approaches are inapplicable to VIs of the so-called second kind, i.e. problems that involve minimization of a functional containing non-differentiable terms. In this article, we evaluate the feasibility of using the reduced basis method (RBM) combined with the empirical interpolation method (EIM) to treat VIs. In the proposed approach, the problem is approximated using a penalty or barrier method, and EIM is then applied to the penalty or barrier term. Numerical examples are presented to assess the performance of the proposed method, in particular the accuracy and computational efficiency of the approximation. Although the numerical examples involve only VIs of the first kind, we also evaluate the feasibility of using the RBM combined with the EIM to treat VIs of the second kind.

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.

Energy-Sustainable Traffic Signal Timings for a Congested Road Network With Heterogeneous Users

This paper proposes a novel model to address the energy-efficient traffic signal timing problem for a congested road network with heterogeneous users. In the proposed model, two types of agents, i.e., the authority and road users, are considered together with the interaction between traffic signal settings and energy policy (e.g., fuel surcharges). To model the route choice behavior of heterogeneous users, a multiclass stochastic traffic network equilibrium problem that considers vehicle delays at signalized intersections and travel demand elasticity is described and formulated as a variational inequality formulation. The authority aims to maximize social welfare of the transportation system by optimizing the traffic signal timings and fuel surcharges. A simulated-annealing-based solution algorithm is developed to solve the proposed model. The findings show that the implementation of the fuel surcharge policy can cause spatial and social inequity issues.

Traffic Network Equilibrium Problems with Capacity Constraints of Arcs and Linear Scalarization Methods

Traffic network equilibrium problems with capacity constraints of arcs are studied. A (weak) vector equilibrium principle with vector-valued cost functions, which are different from the ones in the work of Lin (2010), and three kinds of parametric equilibrium flows are introduced. Some necessary and sufficient conditions for a (weak) vector equilibrium flow to be a parametric equilibrium flow are derived. Relationships between a parametric equilibrium flow and a solution of a scalar variational inequality problem are also discussed. Some examples are given to illustrate our results.

Traffic network equilibrium with capacity constraints and generalized Wardrop equilibrium

The traffic network equilibrium problem with capacity constraints is investigated with the help of generalized Lagrangian theory of variational inequality in our paper. It is proved to be equivalent to the generalized Wardrop equilibrium solution. This result is very useful in transportation analysis since the study in Wardrop equilibrium problem is relatively mature. Furthermore, some new concepts are offered.

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.

Hölder continuity of solutions to elastic traffic network models

This paper aims to study stability and sensitivity analysis for quasi-variational inequalities which model traffic network equilibrium problems with elastic travel demand. In particular, we provide a Holder stability result under parametric perturbations.

REGULARITY RESULTS FOR TIME-DEPENDENT VARIATIONAL AND QUASI-VARIATIONAL INEQUALITIES AND APPLICATION TO THE CALCULATION OF DYNAMIC TRAFFIC NETWORK

The aim of this paper is to consider time-dependent variational and quasi-variational inequalities and to study under which assumptions the continuity of solutions with respect to time can be ensured. Making an appropriate use of the set convergence in Mosco's sense, we are able to prove continuity results for strongly monotone variational and quasi-variational inequalities. The continuity results allow us to provide a discretization procedure for the calculation of solutions to the variational inequalities and, as a consequence, we can solve the time-dependent traffic network equilibrium problem.

On relations between multiclass, multicriteria traffic network equilibrium models and vector variational inequalities

In this paper, relations between vector and weak vector variational inequalities and equilibrium principles for multiclass, multicriteria traffic network equilibrium problem in the fixed demand case and elastic demand case are established, respectively.

A path-based double projection method for solving the asymmetric traffic network equilibrium problem

In this paper we propose a new iterative method for solving the asymmetric traffic equilibrium problem when formulated as a variational inequality whose variables are the path flows. The path formulation leads to a decomposable structure of the constraints set and allows us to obtain highly accurate solutions. The proposed method is a column generation scheme based on a variant of the Khobotov’s extragradient method for solving variational inequalities. Computational experiments have been carried out on several networks of a medium-large scale. The results obtained are promising and show the applicability of the method for solving large-scale equilibrium problems.

Degenerate Time-dependent Variational Inequalities with Applications to Traffic Equilibrium Problems

Abstract The aim of this paper is to study the continuity of the solutions to degenerate time-dependent variational inequalities. In order to obtain the continuity of the solution, a previous continuity result for strongly monotone variational inequalities and an appropriate use of the convergence set in Mosco’s sense play an important role. The continuity result allows us to provide a discretization procedure for the calculation of the solution to the variational inequality which expresses the time-dependent traffic network equilibrium problem.

A modified augmented Lagrangian method for a class of monotone variational inequalities

The augmented Lagrangian method is an attractive approach for solving linearly constrained convex optimization problems (CCOP). The method is iterative and the task in each iteration is to solve a (convex optimization) sub-problem whose dimension is equal to the number of variables. For large problems, solving such a sub-problem can be still very expensive. In this paper, we propose a modified augmented Lagrangian method for a class of monotone variational inequalities which include CCOP. The computational load in each iteration of the proposed method is quite small. In order to demonstrate the ability and efficiency of the proposed method, we present some numerical results for convex nonlinear programming and traffic network equilibrium problems.

The multi-class, multi-criteria traffic network equilibrium and systems optimum problem

It is well known that in the standard traffic network equilibrium model with a single value of time (VOT) for all users, a so-called marginal-cost toll can drive a user equilibrium flow pattern to a system optimum. This result holds when either cost (money) or time units are used in expressing the objective function of the system optimum and the criterion for user equilibrium. This paper examines the multi-criteria or the cost-versus-time network equilibrium and system optimum problem in a network with a discrete set of VOTs for several user classes. Specifically, the following questions are investigated: Are the user-optimal flows dependent upon the unit (time or money) used in measuring the travel disutility in the presence of road pricing? Are there any uniform link tolls across all individuals (link tolls that are identical for all user classes) that can support a multi-class user equilibrium flow pattern as a system optimum when the system objective function is measured by either money or time units? What are the general properties of the valid toll set?

Transportation Network Policy Modeling with Goal Targets and Generalized Penalty Functions

In this paper we develop a transportation network policy model that determines the equilibrium flow pattern and the transportation link tolls or taxes/subsidies in the presence of imposed transportation goals on the links of the network. The penalties (and subsidies) are functions of the deviations from the targets. The governing equilibrium conditions are shown to satisfy a variational inequality problem and existence and uniqueness results are presented using this methodology. A decomposition algorithm is proposed that resolves the problem into series of traffic network equilibrium problems and two simpler subproblems. Convergence results are also given. Finally, the algorithm is applied to numerical examples that illustrate how the model can be utilized for transportation pricing policy, as well as the numerical performance of the algorithm. This work extends recent research in spatial market policy modeling with goal targets on bipartite networks and with fixed penalties.