Model predictive control (MPC) formulations with linear dynamics and quadratic objectives can be solved efficiently by using a primal-dual interior-point framework, with complexity proportional to the length of the horizon. An alternative, which is more able to exploit the similarity of the problems that are solved at each decision point of linear MPC, is to use an active-set approach, in which the MPC problem is viewed as a convex quadratic program that is parametrized by the initial state $x_0$. Another alternative is to identify explicitly polyhedral regions of the space occupied by $x_0$ within which the set of active constraints remains constant, and to pre-calculate solution operators on each of these regions. All these approaches are discussed here.
To appear in "Handbook of Model Predictive Control," 2018.