Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets

TL;DR

This paper develops a data-driven, distributionally robust approach to chance constrained optimization using Wasserstein ambiguity sets, providing convex reformulations and algorithms for different types of constraint functions.

Contribution

It introduces a novel reformulation of DRCCPs with Wasserstein ambiguity sets and develops tractable algorithms for affine, concave, and convex constraint functions.

Findings

Convex reformulation is tractable for affine constraints.

A cutting-surface algorithm converges for concave constraints.

Comparison shows improved feasibility sets over existing methods.

Abstract

We present a data-driven approach for distributionally robust chance constrained optimization problems (DRCCPs). We consider the case where the decision maker has access to a finite number of samples or realizations of the uncertainty. The chance constraint is then required to hold for all distributions that are close to the empirical distribution constructed from the samples (where the distance between two distributions is defined via the Wasserstein metric). We first reformulate DRCCPs under data-driven Wasserstein ambiguity sets and a general class of constraint functions. When the feasibility set of the chance constraint program is replaced by its convex inner approximation, we present a convex reformulation of the program and show its tractability when the constraint function is affine in both the decision variable and the uncertainty. For constraint functions concave in the uncertainty, we show that a cutting-surface algorithm converges to an approximate solution of the convex inner approximation of DRCCPs. Finally, for constraint functions convex in the uncertainty, we compare the feasibility set with other sample-based approaches for chance constrained programs.