A convex programming approach for ridesharing user equilibrium under fixed driver/rider demand

With the proliferation of smartphone-based ridesharing apps around the world, traffic assignment with ridesharing is drawing increasing attention in recent years. A number of ridesharing user equilibrium (RUE) models have been proposed, but most of them are formulated as path-based mixed complementarity problems based on presumed ridesharing price and inconvenience functions, thus are inconvenient to implement in reality. In this study, by redefining the set of feasible driver trajectories and the market equilibrium conditions for ridesharing, we propose an alternative approach to modeling the RUE when the driver- and rider-demand for each OD pair are fixed and given. We show that the resulting RUE conditions can be equivalently transformed into a convex programming problem, and the existence and uniqueness of RUE link flows are guaranteed under mild conditions. The structure of the model is similar to the classic Beckmann's formulation, except for the additional ridesharing demand-supply constraints. So a dual subgradient algorithm with averaging is proposed to solve the problem, and the dual sub-problem can be solved by the Frank-Wolfe method. The algorithm effectively avoids path enumeration, therefore is implementable on large networks. The impact of problem size on the computational efficiency of the algorithm is theoretically analyzed and numerically demonstrated.