This paper considers the problem of evaluating robust control invariant (RCI) sets for linear discrete-time systems subject to state and input constraints as well as additive disturbances. An RCI set has the property that if the system state is inside the set at any one time, then it is guaranteed to remain in the set for all future times using a pre-defined state feedback control law. This problem is important in many control applications. We present a numerically efficient algorithm for the computation of full-complexity polytopic RCI sets. Farkas' Theorem is first used to derive necessary and sufficient conditions for the existence of an admissible polytopic RCI set in the form of nonlinear matrix inequalities. An Elimination Lemma is then used to derive sufficient conditions, in the form of linear matrix inequalities, for the existence of the solution. An optimization algorithm to approximate maximal RCI sets is also proposed. Numerical examples are given to illustrate the effectiveness of the proposed algorithm.