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This paper deals with approximate value iteration (AVI) algorithms applied to discounted dynamic (DP) programming problems. The so-called Bellman residual is shown to be convex in the Banach space of candidate solutions to the DP problem. This fact motivates the introduction of an AVI algorithm with local search that seeks an approximate solution in a lower dimensional space called approximation architecture. The optimality of a point in the approximation architecture is characterized by means of convex optimization concepts and necessary and sufficient conditions to global optimality are derived. To illustrate the method, two examples are presented which were previously explored in the literature.