Accelerated convergence of time‐splitting algorithm by relaxation method

Abstract

The alternating direction method of multipliers (ADMM) is a widely used model predictive control (MPC) acceleration method. It adopts the time-splitting technique, splitting the original problem into independent subproblems. Relaxed ADMM (R-ADMM) is a generalization of ADMM that often achieves faster convergence. However, its parameters must be chosen by an expert user. Besides, the existing convergence proof of R-ADMM adopts a first-order Taylor approximation, which makes the range of relaxation factors conservative. We tackle these weaknesses by giving rigorous evidence and finding the optimal relaxation factor. Firstly, we deduce the convergence of the R-ADMM algorithm, yielding an accurate range of relaxation factors. Then, we analyze the relationship between convergence rate and relaxation factor and conclude that the optimal relaxation factor depends on a recurrence condition. Since splitting introduces the equality constraint, the decoupled states are getting close in the iteration, meeting the recursive requirements and helping find the optimal relaxation factor. Finally, various trajectory tracking tasks are conducted to verify the efficiency of the R-ADMM algorithm. And the simulation results show that the R-ADMM algorithm reduces the number of iterations by 63.7% compared with the ADMM algorithm in the double lane change task.