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The standard version of the policy iteration (PI) algorithm fails forsemicontinuous models, that is, for models with lower semicontinuous one-stepcosts and weakly continuous transition law. This is due to the lack ofcontinuity properties of the discounted cost for stationary policies, thusappearing a measurability problem in the improvement step. The present workproposes an alternative version of PI algorithm which performs an smoothingstep to avoid the measurability problem. Assuming that the model satisfies aLyapunov growth conditions and also some standard continuity-compactnessproperties, it is shown the linear convergence of the policy iterationfunctions to the optimal value function. Strengthening the continuityconditions, in a second result, it is shown that among the improvement policiesthere is one with the best possible improvement and whose cost function iscontinuous.

A note on the policy iteration algorithm for discounted Markov decision processes for a class of semicontinuous models