Linear quadratic model predictive control (MPC) with input constraints leads to an optimization problem that has to be solved at every instant in time. Although there exists computational complexity analysis for current online optimization methods dedicated to MPC, the worst case complexity bound is either hard to compute or far off from the practically observed bound. In this paper we introduce fast gradient methods that allow one to compute a priori the worst case bound required to find a solution with pre-specified accuracy. Both warm- and cold-starting techniques are analyzed and an illustrative example confirms that small, practical bounds can be obtained that together with the algorithmic and numerical simplicity of fast gradient methods allow online optimization at high rates.