Complexity certifications of inexact projection primal gradient method for convex problems: Application to embedded MPC

In this paper we introduce a primal projected gradient method based on inexact projections for solving constrained convex problems. For this algorithm we prove sublinear rate of convergence when applied to problems with objective function being convex and having Lipschitz gradient. At each iteration, our method computes a gradient step towards the solution of the unconstrained problem and then projecting approximately this step onto the feasible set. We recast the inexact projection as approximately solving a best approximation problem for the gradient step until a certain stopping criterion holds. Finally, we show that there are available powerful algorithms, with linear convergence, for computing the inexact projection, such as Dykstra algorithm and alternating direction method of multipliers. Our algorithm is especially useful in embedded model predictive control on hardware with limited computational power, where tight bounds on the computational complexity of the numerical algorithm, used for solving the control problem, are required.