Inverse parametric linear/quadratic programming has recently received increasing attention due to its promising relevance in constrained control design for linear systems. Its aim is to construct an appropriate optimization problem composed of a set of linear constraints and a cost function such that the optimal solution to such a problem, is equivalent to the given continuous piecewise affine function defined over a polyhedral partition. This paper introduces a constructive procedure to find this formulation. The main idea is based on convex lifting. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Furthermore, this convex lifting based method requires at most one supplementary scalar variable. In view of linear model predictive control, it will be shown that any continuous piecewise affine control law can be obtained via a reformulated linear model predictive control problem with the control horizon at most equal to 2 prediction steps.