A contraction approach to periodic optimization problems

Abstract

Consider an infinite horizon, multi-dimensional optimization problem with arbitrary but finite periodicity in discrete time. The problem can be posed as a set of coupled equations. We show that the problem is a special case of a more general class of problems, that the general class has a unique solution, and that the solution can be obtained with the help of a contraction operator. Special cases include the classical Bellman problem and stochastic problem formulations. Thus, we view our approach as an extension of the Bellman problem to the special case of non-autonomy that periodicity represents, and we thereby pave the way for consistent and rigorous treatment of, for example, seasonality in discrete, dynamic optimization. We demonstrate our method in a simple example with periodic variation in the objective function.