Controlled Markov Set-chains with Set-valued Rewards—The Average Case

In this paper, the average case of finite state controlled Markov set-chains with R ^p -set-valued rewards are considered. The optimization is done by a pseudo-order relation on the set of all convex and compact subsets of R ^p induced by a closed convex cone. We introduce a v -step contractive property (minorization condition) for the average case and by use of this method the average expected reward set from a periodic policy is characterized. And, applying the scalarization technique, a Pareto optimal policy is obtained. Also, a numerical example is given to comprehend our results.