An alternating direction method of multipliers for solving user equilibrium problem

• Published in: European journal of operational research Vol. 310; no. 3; pp. 1072 - 1084
• Main Authors: Liu, Zhiyuan, Chen, Xinyuan, Hu, Jintao, Wang, Shuaian, Zhang, Kai, Zhang, Honggang
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2023
• Subjects: Alternating direction method of multipliers; Edge-coloring problem; Parallel computing; Traffic assignment; User equilibrium; Alternating direction method of multipliers; User equilibrium; Parallel computing; Traffic assignment; Edge-coloring problem
• ISSN: 0377-2217
• DOI: 10.1016/j.ejor.2023.04.008

•We propose an alternating direction method of multipliers (ADMM) to address the user equilibrium problem.•We propose an almost-optiaml edge coloring algorithm to group transport network links into different blocks.•The proposed ADMM algorithm enables parallel computing which expedites the convergency rate. This paper introduces a new parallel computing algorithm to address the user equilibrium (UE) problem. Searching for efficient solution algorithms for UE has been a recurring study subject in transportation research and has attracted much attention in past decades. Existing solution algorithms can be classified into three categories: link-based, path-based, and origin-based. This paper introduces an alternating direction method of multipliers (ADMM) algorithm that is different from these categories. Based on the origin-based formulation of UE problem, an equivalent problem is proposed which eliminates the flow conservation conditions through the augmented Lagrangian function. In order to make use of the ADMM, the network links should be grouped into different blocks, where the links in the same block are disconnected. This link grouping problem falls into the category of edge-coloring problem in graph theory, and it follows the Vizing theorem. A novel approach is developed for the link grouping problem. For links in the same block, we have a separable subproblem, which is solved in parallel by the gradient projection algorithm. Numerical experiments are conducted to validate the proposed algorithm, which shows its computation efficiency.