  Recently there has been an increasing interest in primal-dual methods for
model predictive control (MPC), which require minimizing the (augmented)
Lagrangian at each iteration. We propose a novel first order primal-dual
method, termed \emph{proportional-integral projected gradient method}, for MPC
where the underlying finite horizon optimal control problem has both state and
input constraints. Instead of minimizing the (augmented) Lagrangian, each
iteration of our method only computes a single projection onto the state and
input constraint set. Our method ensures that, along a sequence of averaged
iterates, both the distance to optimum and the constraint violation converge to
zero at a rate of \(O(1/k)\) if the objective function is convex, where \(k\)
is the iteration number. If the objective function is strongly convex, this
rate can be improved to \(O(1/k^2)\) for the distance to optimum and
\(O(1/k^3)\) for the constraint violation. We compare our method against
existing methods via a trajectory-planning example with convexified
keep-out-zone constraints.
