Scaling limits of multi-period distributionally robust optimization problems

TL;DR

This paper analyzes the behavior of multi-period distributionally robust optimization problems as the period length shrinks, revealing a connection to nonlinear PDEs and continuous-time robust optimization.

Contribution

It introduces a semigroup approach to characterize the scaling limit of multi-period DRO and identifies its infinitesimal generator, linking discrete robust problems to continuous-time PDEs.

Findings

The scaling limit forms a strongly continuous monotone semigroup.

The infinitesimal generator combines non-robust and Wasserstein-induced perturbations.

Viscosity solutions of the PDE match continuous-time robust optimization values.

Abstract

We examine the scaling limit of multi-period distributionally robust optimization (DRO) problems via a semigroup approach. Each period involves a worst-case maximization over distributions in a Wasserstein ball around the transition probability of a reference process with radius proportional to the length of the period, and the multi-period DRO problem arises through its sequential composition. We show that the scaling limit of the multi-period DRO, as the length of each period tends to zero, is a strongly continuous monotone semigroup on $\mathrm{C_b}$. Furthermore, we show that its infinitesimal generator is equal to the generator associated with the non-robust scaling limit plus an additional perturbation term induced by the Wasserstein uncertainty. As an application, we show that when the reference process follows an It\^o process, the viscosity solution of the associated nonlinear PDE coincides with the value of continuous-time robust optimization problems under parametric uncertainty.