The principle of optimality in dynamic programming: A pedagogical note

The principle of optimality is a fundamental aspect of dynamic programming, which states that the optimal solution to a dynamic optimization problem can be found by combining the optimal solutions to its sub-problems. While this principle is generally applicable, it is often only taught for problems with finite or countable state spaces in order to sidestep measure-theoretic complexities. Therefore, it cannot be applied to classic models such as inventory management and dynamic pricing models that have continuous state spaces, and students may not be aware of the possible challenges involved in studying dynamic programming models with general state spaces. To address this, we provide conditions and a self-contained simple proof that establish when the principle of optimality for discounted dynamic programming is valid. These conditions shed light on the difficulties that may arise in the general state space case. We provide examples from the literature that include the relatively involved case of universally measurable dynamic programming and the simple case of finite dynamic programming where our main result can be applied to show that the principle of optimality holds.