Summary: We study two outer approximation schemes, applied to the variational inequality problem in reflexive Banach spaces. First we propose a generic outer approximation scheme, and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem. As is standard in this setting, boundedness and optimality of weak limit points are proved to hold under two alternative conditions: (i) boundedness of the feasible set, or (ii) coerciveness of the operator. To develop a convergence analysis where (i) and (ii) do not hold, we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator. Under conditions alternative to both (i) and (ii), we obtain standard convergence results. Furthermore, when the space is uniformly convex, we establish full strong convergence of the second scheme to a solution.