Compactness of the space of non-randomized policies in countable-state sequential decision processes

摘要

For sequential decision processes with countable state spaces, we prove compactness of the set of strategic measures corresponding to nonrandomized policies. For the Borel state case, this set may not be compact (Piunovskiy, Optimal control of random sequences in problems with constraints. Kluwer, Boston, p. 170, 1997) in spite of compactness of the set of strategic measures corresponding to all policies (Schal, On dynamic programming: compactness of the space of policies. Stoch Processes Appl 3(4):345-364, 1975b; Balder, On compactness of the space of policies in stochastic dynamic programming. Stoch Processes Appl 32(1):141-150, 1989). We use the compactness result from this paper to show the existence of optimal policies for countable-state constrained optimization of expected discounted and nonpositive rewards, when the optimality is considered within the class of nonrandomized policies. This paper also studies the convergence of a value-iteration algorithm for such constrained problems.