Dynamic Programs on Partially Ordered Sets

TL;DR

This paper presents a unified order-theoretic framework for dynamic programming, encompassing various applications from traditional to advanced nonlinear and probabilistic methods, leading to new optimality and algorithmic insights.

Contribution

It introduces a novel framework representing dynamic programs via operators on partially ordered sets, unifying diverse applications and deriving new theoretical results.

Findings

Unified order-theoretic framework for dynamic programming

Application to nonlinear recursive preferences and distributional methods

New optimality and algorithmic results for specific applications

Abstract

We introduce a framework that represents a dynamic program as a family of operators acting on a partially ordered set. We provide an optimality theory based only on order-theoretic assumptions and show how applications across almost all subfields of dynamic programming fit into this framework. These range from traditional dynamic programs to those involving nonlinear recursive preferences, desire for robustness, function approximation, Monte Carlo sampling and distributional dynamic programs. We apply the framework to establish new optimality and algorithmic results for specific applications.