Iteration complexity of an inexact augmented Lagrangian method for constrained MPC

In this paper we discuss the iteration complexity certification for solving constrained MPC problems for linear systems using an inexact augmented Lagrangian approach. We solve the augmented dual problem that arises from Lagrange relaxation of the linear constraints coming from the dynamics with the dual gradient method. Since the exact solution of the primal augmented Lagrange problem is usually impossible to compute in practical situations, we consider an inexact version of the dual gradient method. We discuss the relation between the inner and the outer accuracy of the primal and dual problem and we derive lower bounds on both primal and dual gap but also primal feasibility. We prove that we can obtain an o accurate solution in terms of the primal optimality and feasibility in at most O(1/ϵ) iterations, provided that the inner problems are solved with accuracy ε².