Inverse parametric convex programming problems via convex liftings

The present paper introduces a procedure to recover an inverse parametric linear or quadratic programming problem from a given polyhedral partition over which a continuous piecewise affine function is defined. The solution to the resulting parametric linear problem is exactly the initial piecewise affine function over the given original parameter space partition. We provide sufficient conditions for the existence of solutions for such inverse problems. Furthermore, the constructive procedure proposed here requires at most one supplementary variable in the vector of optimization arguments. The principle of this method builds upon an inverse map to the orthogonal projection, known as a convex lifting. Finally, we show that the theoretical results has a practical interest in Model Predictive Control (MPC) design. It is shown that any linear Model Predictive Controller can be obtained through a reformulated MPC problem with control horizon equal to two prediction steps.