Technical Note—On the Strength of Relaxations of Weakly Coupled Stochastic Dynamic Programs

Many sequential decision problems have a weakly coupled structure in that a set of linking constraints couples an otherwise independent collection of subproblems. This structure arises in a wide variety of applications, such as network revenue management, online advertising, assortment planning, interactive marketing, optimization of power systems, and multilocation inventory management to name only a few. Such problems can be modeled as dynamic programs but are quite difficult to solve. Two widely studied approximation methods are approximate linear programs, which involve finding a best approximation of total value that is additive across the subsystems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well known that both of these approaches provide upper bounds to the optimal value, and the approximate linear programming approach is a better bound but also, more difficult to compute. In this paper, we provide a detailed theoretical analysis of these two approximations and show that, under fairly broad conditions, these two approximations lead to upper bounds that are very close and often identical. Our theory suggests that, between these two approximations, Lagrangian relaxations should usually be the preferred choice for researchers studying applications involving weakly coupled dynamic programs.