Abstract
In this paper, we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank–Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows in the stable dynamics model, which provides us with a computable duality gap.
Highlights
The Beckmann model for searching static traffic equilibria in road networks is among the most widely used models by transportation planners [1,2]
We propose a novel way to reconstruct admissible flows in the stable dynamics model and a novel computable duality gap, which can be used in a stopping criterion
We considered several primal-dual subgradient methods for finding equilibria in the stable dynamics and the Beckmann models
Summary
The Beckmann model for searching static traffic equilibria in road networks is among the most widely used models by transportation planners [1,2]. The equilibria found are practical for evaluating the network efficiency and distribution of business centers and residential areas, and establishing urban development plans, etc This model introduces a cost function on every link of a transportation network, which defines a dependence of the travel cost on the flow along the link. In the Beckmann model, equilibria are as follows: for all three cases, only the upper route is used, and the equilibrium travel times are approximately 0.5, 0.6, and 0.9 h, respectively. We compare several primal-dual gradient methods for searching equilibria in both the Beckmann and the stable dynamics models, namely, the universal gradient method (UGM) [12], the universal method of similar triangles (UMST) [13], and the method of weighted dual averages (WDA) [14].
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