Existence of efficient solutions in infinite horizon optimization under continuous and discrete controls

We consider a general deterministic infinite horizon optimization problem over discrete time with time-varying, i.e., non-stationary, data. Our formulation requires only that action spaces be compact, including both continuous and discrete controls. In the event that all total costs diverge, i.e., no least total cost optimum exists, we investigate the existence of efficient optima. (An infinite horizon feasible solution is efficient if it is optimal to each of the states through which it passes.) We show that the mapping from controls to states (i.e. state transition function) being open is a sufficient condition for existence of efficient solutions. In this event, we also give a necessary and sufficient condition for there to exist a unique efficient optimum. Our results are then applied to an infinite horizon production planning problem with no backlogging.