Computational complexity certification for dual gradient method: Application to embedded MPC

In this paper we analyze the computational complexity of the dual gradient method for solving linearly constrained convex problems. When it is difficult to project on the primal feasible set described by linear constraints, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply the dual gradient algorithm for solving the corresponding dual. We give a unified convergence rate analysis for the dual gradient algorithm: we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions. Our analysis relies on the Lipschitz property of the dual function or an error bound property. Furthermore, the iteration complexity analysis is based on two types of approximate primal solutions: an average primal sequence or the last primal iterate sequence. We also discuss complexity certifications and implementation aspects of the dual gradient algorithm on constrained MPC problems for embedded linear systems.