Optimality in Multi-stage Operations with Asymptotically Vanishing Cost

This paper sets out a framework for discussing operations whose cost can be made to approach zero by subdividing the operation into an increasing number (K) of stages. Examples of such processes include what thermodynamics books call quasistatic processes with wasted work as the cost. More generically, they include processes taking place with near-equilibrium conditions between the participants. For any fixed value of K, there are always many ways to carry out the subdivision. This paper addresses some questions related to the asymptotic (large K) behavior of the minimum total cost. In particular, we show that corresponding points in optimal subdivisions for objectives with the same quadratic part differ by O(1/K), while the values of the objective functions differ by O(1/K2). Our main result is the construction of a geometrically motivated near-optimal partition scheme whose total cost (for each K) differs from the true minimum by O(1/K3). The research is motivated by recent efforts in the analysis of entropy production minimization for thermodynamic processes. In that context, our result shows that the equal thermodynamic distance subdivision will come within O(1/K3) of the true minimum entropy production.