Dynamic Programming Deconstructed: : Transformations of the Bellman Equation and Computational Efficiency

Transforming Dynamic Optimization Problems for Enhanced Computational Efficiency

Dynamic programming is one of the core algorithms for solving optimization problems in operations research, economics, finance, computer science, and numerous other fields. Although the theory itself is relatively straightforward, implementation can be computationally expensive, inhibiting application to interesting and realistic problems. This research considers a variety of transformations that can be applied to optimization problems and the key equations associated with the dynamic programming algorithm. It investigates the range of possible transformations, clarifies their links with optimality, and applies specific transformations to generate large speed gains in high-dimensional problems. Several economic applications are considered.

Some approaches to solving challenging dynamic programming problems, such as Q-learning, begin by transforming the Bellman equation into an alternative functional equation to open up a new line of attack. Our paper studies this idea systematically with a focus on boosting computational efficiency. We provide a characterization of the set of valid transformations of the Bellman equation, for which validity means that the transformed Bellman equation maintains the link to optimality held by the original Bellman equation. We then examine the solutions of the transformed Bellman equations and analyze correspondingly transformed versions of the algorithms used to solve for optimal policies. These investigations yield new approaches to a variety of discrete time dynamic programming problems, including those with features such as recursive preferences or desire for robustness. Increased computational efficiency is demonstrated via time complexity arguments and numerical experiments.