Dynamic Programming in Ordered Vector Space
Nisha Peng, John Stachurski
TL;DR
This paper develops an abstract dynamic programming framework in ordered vector spaces, showing how lattice and algebraic structure yield sharper fixed-point results and stronger optimality guarantees than in purely abstract poset settings. By leveraging concavity, contraction-like properties, and affine operator structure, the authors obtain existence, uniqueness, and convergence results for fixed points and for standard DP algorithms such as VFI, OPI, and HPI. The framework is bridged to concrete applications, including data valuation, risk-sensitive preferences, quantile learning, and nonlinear discounting, demonstrating regularity, convergence, and practical interpretability of the results. The approach enhances testability and applicability of abstract DP theory across diverse economic and AI contexts, providing a robust toolkit for both theory and applied optimization.
Abstract
Recent approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered vector space. The advantage of working in this setting is that ordered vector spaces have well integrated algebric and order structure, which leads to sharper fixed point results. These fixed point results can then be exploited to obtain strong optimality properties. We illustrate our results through a range of applications, including new findings for several useful models.