A novel framework of the alternating direction method of multipliers with application to traffic assignment problem

This paper proposes a novel algorithmic framework to enhance the convergence efficiency of the alternating direction method of multipliers (ADMM) by incorporating the successive over relaxation (SOR) splitting method. The proposed framework holds applicability across various research fields for improving convergence efficiency. Currently, there exist two main methods for decomposing the separate optimization problems: Gauss-Seidel (GS) and Jacobi methods. The SOR method introduced in this paper offers a more efficient alternative. Following the original ADMM algorithm’s framework, we provide a detailed procedure for incorporating the SOR method into the ADMM framework in place of the GS splitting method. This development gives rise to a new method called ADMM-SOR, and then we apply this newly proposed algorithm to solve the deterministic user equilibrium (DUE) problem. Subsequently, to ensure the reliability of the proposed algorithm, we rigorously prove its convergence by leveraging some properties of variational inequalities. Additionally, the impact of the relaxation factor on the efficiency of the ADMM-SOR method is conducted, and we also explore a novel method to self-adjust the relaxation factor during each iteration. The new algorithm is verified based on numerical experiments, revealing that the novel ADMM-SOR framework achieves faster convergence in comparison to the original one, all the while maintaining exceptional parallel performance.