Rate analysis of inexact dual gradient methods: Application to embedded and distributed MPC

In this talk we provide rate analysis and computational complexity certification of inexact dual gradient methods for solving convex optimization problems with complicated constraints. We solve the (augmented) Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the (augmented)Lagrangian primal problem is hard to compute in practice, we solve this problem up to some given inner accuracy. We derive relations between the inner and the outer accuracy of the primal and dual problems and we give a full convergence rate analysis for both gradient and fast gradient algorithms. We provide estimates on the primal and dual suboptimality and on primal feasibility violation of the generated approximate primal and dual solutions. Our analysis relies on the Lipschitz property of the dual function and on inexact dual gradients. We also discuss implementation aspects of the proposed algorithms on constrained model predictive control problems for embedded linear systems or network systems.