— Constructive solution to inverse parametric lin-ear/quadratic programming problems has recently been investigated and shown to be solvable via convex liftings [15], [14]. These results were stated and solved starting from polytopic partitions of a polytope in the parameter space. Therefore, the case of polyhedral partitions of unbounded polyhedra, was not handled by this method and deserves a complete characterization to address the general inverse optimality problem. This paper has as main objective to overcome the unboundedness limitation of the given polyhedral partition and to extend the constructive solution put forward in [14] for this omitted case.