This paper presents a new form of robust Model Predictive Control (MPC) using constraint tightening, where the degree of tightening is a convex function of the feedback parameters. The proposed approach is shown to be able to represent a strictly larger class of feedback policies when compared to previous algorithms. Further analytical results provide a) necessary and sufficient conditions on the choice of feedback parameters for the existence of a nonempty output constraint set; and b) a sufficient condition for the existence of a nonempty robust invariant set. Combined with the convex parametrization, this enables an off-line linear optimization to determine the feedback policy that can tolerate much stronger disturbances while robustly satisfying the constraints. Simulation results are presented to highlight the advantages of the new control algorithm.