Model predictive control (MPC) is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables. To circumvent the main drawback of MPC, i.e., the need to solve a Quadratic Program (QP) on line to compute the control action, explicit MPC was proposed in the past to precompute the control law off line using multiparametric QP (mpQP). The resulting form of the MPC law is piecewise affine, which is extremely easy to code, can be computed online by simple arithmetic operations, and requires a maximum number of iterations that can be exactly determined a priori. On the other hand, the offline computations to solve the mpQP problem require detecting emptiness, full-dimensionality, and minimal hyperplane representations of polyhedra, and other computational geometric operations. While most of the existing methods solve such operations via linear programming, the approach proposed in this paper relies on a nonnegative least squares (NNLS) solver that is very simple to code and fast to execute. In addition, the new approach exploits QP duality to identify and construct critical regions and to handle degeneracy issues.