Average optimality for Markov decision processes in borel spaces: a new condition and approach

In this paper we study discrete-time Markov decision processes with Borel state and action spaces. The criterion is to minimize average expected costs, and the costs may have neither upper nor lower bounds. We first provide two average optimality inequalities of opposing directions and give conditions for the existence of solutions to them. Then, using the two inequalities, we ensure the existence of an average optimal (deterministic) stationary policy under additional continuity-compactness assumptions. Our conditions are slightly weaker than those in the previous literature. Also, some new sufficient conditions for the existence of an average optimal stationary policy are imposed on the primitive data of the model. Moreover, our approach is slightly different from the well-known ‘optimality inequality approach’ widely used in Markov decision processes. Finally, we illustrate our results in two examples.

Average optimality for Markov decision processes in borel spaces: a new condition and approach

In this paper we study discrete-time Markov decision processes with Borel state and action spaces. The criterion is to minimize average expected costs, and the costs may have neither upper nor lower bounds. We first provide two average optimality inequalities of opposing directions and give conditions for the existence of solutions to them. Then, using the two inequalities, we ensure the existence of an average optimal (deterministic) stationary policy under additional continuity-compactness assumptions. Our conditions are slightly weaker than those in the previous literature. Also, some new sufficient conditions for the existence of an average optimal stationary policy are imposed on the primitive data of the model. Moreover, our approach is slightly different from the well-known ‘optimality inequality approach’ widely used in Markov decision processes. Finally, we illustrate our results in two examples.

The Average Cost Optimality Equation and Critical Number Policies

We consider a Markov decision chain with countable state space, finite action sets, and nonnegative costs. Conditions for the average cost optimality inequality to be an equality are derived. This extends work of Cavazos-Cadena [8]. It is shown that an optimal stationary policy must satisfy the optimality equation at all positive recurrent states. Structural results on the chain induced by an optimal stationary policy are derived. The results are employed in two examples to prove that any optimal stationary policy must be of critical number form.