Policy Iteration in Finite Templates Domain

We prove in this paper that policy iteration can be generally defined in finite domain of templates using Lagrange duality. Such policy iteration algorithm converges to a fixed point when for very simple technique condition holds. This fixed point furnishes a safe over-approximation of the set of reachable values taken by the variables of a program. We prove also that policy iteration can be easily initialised for one single loop programs when templates are correctly chosen.