The paper deals with the control of linear discrete-time systems in presence of non-convex state-space constraints, understood as obstacles for the closed-loop trajectories. These interdicted regions can include the origin and as a consequence, the natural limit behavior has to be replaced by a new admissible equilibrium point, possibly by activating a subset of constraints. It will be shown that a predictive control strategy can deal, by its generic formulation, with such problems and several refinements will be pointed out with respect to the dual-mode principles and the numerical formulations of the optimization problems. Furthermore, the second goal of the paper is to present an interpolation-based procedure as a reduced-complexity alternative to the classical receding horizon optimization. The basic elements for the establishment of such a control scheme are the boundary of the closed convex sets in the state-space and the associated vertex controller.