A variational inequality formulation for stochastic user equilibrium with a bounded choice set

This study develops a variational inequality (VI) formulation for the stochastic user equilibrium with a bounded choice set, known as the bounded choice model (BCM). A novel mapping is devised to facilitate the development of the VI formulation. The VI formulation offers a new avenue for analysing and solving the BCM. It is shown that all paths in the considered choice set should have an equal mapping at the optimal solution, which resembles the equilibrated travel cost in the user equilibrium (UE). The new formulation has two algorithmic benefits. First, an algorithm that solves a VI problem (i.e., the extragradient method) can outperform the method of successive averages (MSA) in large networks under a UE-type convergent measurement. Second, it is possible to avoid the column reduction procedure when finding the choice set that satisfies the BCM requirement because a projection-type algorithm that solves the VI formulation can assign zero flow to a path. This is more efficient than the MSA, which uses diminishing step sizes to update solutions over iterations, leading to a sublinear rate of convergence and small residual flows on paths that should carry zero flows. Numerical experiments are conducted to demonstrate the properties of the model and compare the performance of different algorithms using the Sioux Falls, Eastern Massachusetts, and Anaheim networks.