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This paper develops a parameterized tube model predictive control (MPC) synthesis method. The most relevant novel feature of our proposal is the online use of a single tractable linear program that optimizes parameterized, Minkowski decomposable, state and control tubes and an associated, fully separable, nonlinear, control policy. The induced control policy enjoys a higher degree of nonlinearity than existing tube MPC and robust MPC using disturbance affine control policy. Our proposal offers greater generality than the state of the art robust MPC methods. It is conjectured, and also established in three cases, that our proposal is equivalent, feasibility-wise, to dynamic programming (DP). It is also shown that, under natural assumptions, our method is computationally efficient while it possesses rather strong system theoretic properties.