  Policy iteration and value iteration are at the core of many (approximate)
dynamic programming methods. For Markov Decision Processes with finite state
and action spaces, we show that they are instances of semismooth Newton-type
methods to solve the Bellman equation. In particular, we prove that policy
iteration is equivalent to the exact semismooth Newton method and enjoys local
quadratic convergence rate. This finding is corroborated by extensive numerical
evidence in the fields of control and operations research, which confirms that
policy iteration generally requires few iterations to achieve convergence even
when the number of policies is vast. We then show that value iteration is an
instance of the fixed-point iteration method. In this spirit, we develop a
novel locally accelerated version of value iteration with global convergence
guarantees and negligible extra computational costs.
