Infinite horizon optimization

The authors consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces. Examples include equipment replacement, production planning and, more generally, infinite stage mathematical programs. It is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity. Moreover, the existence of a unique algorithmically optimal (i.e. accumulation point) solution is shown to be a necessary and sufficient condition for convergence in the product topology (i.e. policy convergence) of all finite horizon optima. Under the weaker condition of Hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima, the authors show how to force policy convergence through a natural tie-breaking rule. Finally, a forward algorithm is presented which, in the presence of a unique infinite horizon optimum, is guaranteed to converge.