Abstract A convergence analysis of the alternating direction method of multipliers (ADMM) for linear model predictive control (MPC) problems with regularization terms is addressed here. Compared with conventional results, this paper focuses on the dynamical structure of the ADMM and derives the linear convergence of the state variables of the ADMM dynamics.In on‐line optimization problems associated with MPC problems, because the computation time is a significant issue, a hot‐start technique in which an initial point of the algorithm is set as the convergence point for the previous optimization problem is widely employed to improve the convergence property. Here, by utilizing the proposed convergence analysis framework, the effectiveness of the hot‐start is explored, and the upper bounds of the number of iterations to guarantee the required accuracy of an obtained iterative solution are deduced. The proposed hot‐start framework derives a criterion for choosing the penalty parameter of the ADMM in MPC problems and facilitates effective iteration, which is validated in a numerical example.