We investigate direct numerical solvers in linear model predictive control, where the prediction model is given by linear systems subject to linear inequality constraints on the state and the input, and the performance index is convex and quadratic. The inequality constraints are treated by the primal-dual interior-point method. We propose a novel direct solver based on the augmented Lagrangian regularization of a reduced Hessian. The new solver has the same arithmetic complexity as the factorized Riccati recursion. The direct solver can be implemented in terms of BLAS3 matrix operations.