Consistency of Distributionally Robust Risk- and Chance-Constrained Optimization under Wasserstein Ambiguity Sets

TL;DR

This paper investigates the stability of distributionally robust stochastic optimization problems with risk and chance constraints under Wasserstein ambiguity sets, showing convergence of solutions as sample size grows.

Contribution

It proves that the optimal values and solutions of distributionally robust risk and chance-constrained problems converge to the original problem's solutions with increasing data under certain conditions.

Findings

Optimal values converge to original problem as sample size increases

Solutions stabilize under Wasserstein ambiguity sets

Provides theoretical guarantees for data-driven risk optimization

Abstract

We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of the distributionally robust versions of these problems converge to the respective quantities of the original problems, as the sample size increases.