  The standard version of the policy iteration (PI) algorithm fails for
semicontinuous models, that is, for models with lower semicontinuous one-step
costs and weakly continuous transition law. This is due to the lack of
continuity properties of the discounted cost for stationary policies, thus
appearing a measurability problem in the improvement step. The present work
proposes an alternative version of PI algorithm which performs an smoothing
step to avoid the measurability problem. Assuming that the model satisfies a
Lyapunov growth conditions and also some standard continuity-compactness
properties, it is shown the linear convergence of the policy iteration
functions to the optimal value function. Strengthening the continuity
conditions, in a second result, it is shown that among the improvement policies
there is one with the best possible improvement and whose cost function is
continuous.
