Traffic Network Equilibrium Problem Research Articles

Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric Polyhedral Constraint

It is proved that the metric projection from a point onto a moving polyhedron is Lipschitz continuous with respect to the perturbations on the right-hand sides of the linear inequalities defining the polyhedron. The property leads to a simple sufficient condition for Lipschitz continuity of a locally unique solution of parametric variational inequalities with a moving polyhedral constraint set. Applications of these results to traffic network equilibrium problems are given in detail.

A differential game approach to the dynamic mixed behavior traffic network equilibrium problem

The dynamic mixed behavior traffic network equilibrium model is formulated as a noncooperative N-person nonzero-sum differential game under the open-loop information structure. A simple network is considered where one origin-destination pair is connected by parallel arcs and two types of players - User Equilibrium (UE) and Cournot-Nash (C-N) - interact through the congestion phenomenon. Each of UE and C-N players attempts to achieve its own prescribed objective by making a continuum of simultaneous decisions of departure time, route, and departure flow rate over a fixed time interval. The necessary and sufficient conditions are derived and given economic interpretation as a dynamic game theoretic generalization of the mixed behavior traffic network equilibrium principle which requires equilibration of average costs for UE players and equilibration of marginal costs for C-N players. An approximate iterative algorithm is proposed for solving the model in discrete time, which makes use of the augmented Lagrangian method and the gradient method. A numerical example is presented and future extensions of the model and the algorithm are also discussed.

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.

A network model of migration equilibrium with movement costs

In this paper we introduce a new network equilibrium model of human migration in the presence of movement costs. The equilibrium conditions state that migration continues until the net gain in utility between the destination locations and each particular origin location minus the movement cost is equal, maximal and nonnegative, if there is a positive outward flow from that origin to the particular destinations. We then consider the case of linear utility functions and fixed movement costs and demonstrate that the equilibrium conditions can be reformulated as the solution to an equivalent quadratic programming problem. This approach is in line with the classical approaches to both traffic network equilibrium and spatial price equilibrium problems. We then present an equilibration algorithm which explicitly exploits the special structure of the underlying network. The algorithm proceeds from location to location, at each step equilibrating the migration flow out of a location. The numerical results demonstrate the robustness and efficacy of this approach on large-scale migration network equilibrium problems.

An Application of Fixed Point Algorithm to Traffic Network Equilibrium Problem

An Application of Fixed Point Algorithm to Traffic Network Equilibrium Problem

Numerical experience with diagonalization algorithms for asymmetric demand traffic assignment

Abstract This paper reviews the concept of a diagonalization algorithm for use in solving traffic network equilibrium problems for which the arc cost and/or the origin-destination travel demand functions are asymmetric. Such functions are known to occur in realistic settings involving multiple modes or users. The computational performance of this algorithm for different degrees of travel demand asymmetry is then explored by a detailed numerical experiment since no previous results of this type have been reported. It is found that, through the use of progressive stopping tolerances, the impact of high degrees of travel demand function asymmetry on the computational burden associated with finding a traffic network equilibrium may be mitigated; in effect equilibrium problems with high degrees of demand asymmetry are little more difficult to solve than perfectly symmetric problems.