Abstract
We consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time‐varying, that is, nonstationary, data. The task is to choose a criterion that is neither too overselective, so that no policy is optimal, nor too underselective, so that most policies are optimal. We contrast and compare the following optimality criteria: strong, overtaking, weakly overtaking, efficient, and average. However, our focus is on the optimality criterion of efficiency. (A solution is efficient if it is optimal to each of the states through which it passes.) Under mild regularity conditions, we show that efficient solutions always exist and thus are not overselective. As to underselectivity, we provide weak state reachability conditions which assure that every efficient solution is also average optimal, thus providing a sufficient condition for average optima to exist. Our main result concerns the case where the discounted per‐period costs converge to zero, while the discounted total costs diverge to infinity. Under the assumption that we can reach from any feasible state any feasible sequence of states in bounded time, we show that every efficient solution is also overtaking, thus providing a sufficient condition for overtaking optima to exist.
Highlights
The problem of optimally selecting a sequence of decisions over an infinite horizon is complicated by the criterion issue of imposing preferences over the collection of associated cost streams
We will see that the efficiency criterion is not overselective, since the existence of efficient solutions is assured by relatively mild topological conditions. (We give a reasonable sufficient condition for efficient optimal solutions to exist in our discrete-time, nonstationary, continuous state and control framework.) Nor is it underselective, since such a strategy must be optimal to every state attained along that path
In the presence of average cost reachability, we show that efficient solutions are average optimal (Theorem 4.3)
Summary
The problem of optimally selecting a sequence of decisions over an infinite horizon is complicated by the criterion issue of imposing preferences over the collection of associated cost streams. (We give a reasonable sufficient condition for efficient optimal solutions to exist in our discrete-time, nonstationary, continuous state and control framework.) Nor is it underselective, since such a strategy must be optimal to every state attained along that path. In the discrete action setting of Schochetman and Smith [18], it was shown that, under a (rather strong) state-reachability condition, every efficient solution is average optimal We weaken this state-reachability condition and extend this result to the case of continuous states and controls. We present a mild condition which is sufficient to guarantee the existence of efficient solutions (Theorem 3.4) It is known (Halkin [9]), that weakly overtaking optima are efficient for continuous-time and vector states. They do not consider the average optimality criterion at all there
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