In model predictive control an optimization problem has to be solved at each sampling instant. The objective in this article is to derive efficient methods to solve this optimization problem. The approach taken is to use interior point optimization methods. The model predictive control problem considered here has a quadratic objective and constraints which can be both linear and quadratic. The key to an efficient implementation is to rewrite the optimization problem as a second order cone program. To solve this optimization problem a feasible primal-dual interior point method is employed. By using a feasible method it is possible to determine when the problem is feasible or not by formalizing the search for strictly feasible initial points as yet another primal-dual interior point problem. There are several different ways to rewrite the optimization problem as a second order cone program. However, done carefully, it is possible to use very efficient scalings as well as Riccati recursions for computing the search directions. The use of Riccati recursions makes the computational complexity grow at most as O(N-3/2) with the time horizon, compared to O(N-3) for more standard implementations.