An improved alternating direction method of multipliers for solving deterministic user equilibrium

An improved alternating direction method of multipliers (iADMM) algorithm is proposed for solving the deterministic user equilibrium (DUE) problem. The iADMM algorithm introduces a key improvement wherein the primal variables can be updated sequentially, possibly multiple times, before updating the dual variables. Subsequently, a sensitivity analysis method is applied to optimise the update frequency of primal variable, and two indices are introduced to evaluate the performance of various policies. Additionally, eliminating equality constraints through the augmented Lagrangian function makes it challenging to completely satisfy flow conservation conditions, resulting in the presence of primal residual. To address this issue, two schemes are proposed to handle the primal residual, involving modifications to the primal variables based on iterative messages. Numerical experiments highlight that the convergence of ADMM algorithm is significantly influenced by the update frequency of primal variable, and addressing the primal residual after each cycle enhances the convergence of ADMM algorithm.