This paper focuses on the (re-)construction of the optimal solution for the multi-parameter quadratic programming (mpQP) problems. Optimization problems of this nature are widely employed in the formulation of modelbased predictive controllers (MPC) for discrete linear systems, wherein input and state constraints are imposed. This study examines the geometric characteristics of the explicit solution of an mpQP problem and introduces a novel convex-concave lifting technique to synthesize an equivalent mpQP problem. Whenever the solution corresponds to a PWA function, the present approach maintains the structure and control laws for the original systems. A new (less complex) cost function and a corresponding feasible domain are constructed through lifting for the equivalent optimization problem. The effectiveness of this strategy is demonstrated through an illustrative example.