Using affine policies to reformulate two-stage Wasserstein distributionally robust linear programs to be independent of sample size

TL;DR

This paper introduces a reformulation of two-stage Wasserstein distributionally robust linear programs using affine policies, making the problem independent of sample size and more computationally feasible.

Contribution

The authors develop a scale-independent reformulation of 2-DRLPs with affine policies, enabling efficient solutions regardless of sample size, and propose a probabilistic method to reduce conservativeness.

Findings

Reformulated 2-DRLP is scalable and independent of sample size.

The proposed model outperforms existing methods in out-of-sample performance.

Numerical experiments confirm computational efficiency and effectiveness in power system applications.

Abstract

Intensively studied in theory as a promising data-driven tool for decision-making under ambiguity, two-stage distributionally robust optimization (DRO) problems over Wasserstein balls are not necessarily easy to solve in practice. This is partly due to large sample size. In this article, we study a generic two-stage distributionally robust linear program (2-DRLP) over a 1-Wasserstein ball using an affine policy. The 2-DRLP has right-hand-side uncertainty with a rectangular support. Our main contribution is to show that the 2-DRLP problem has a tractable reformulation with a scale independent of sample size. The reformulated problem can be solved within a pre-defined optimality tolerance using robust optimization techniques. To reduce the inevitable conservativeness of the affine policy while preserving independence of sample size, we further develop a method for constructing an uncertainty set with a probabilistic guarantee over which the Wasserstein ball is re-defined. As an application, we present a novel unit commitment model for power systems under uncertainty of renewable energy generation to examine the effectiveness of the proposed 2-DRLP technique. Extensive numerical experiments demonstrate that our model leads to better out-of-sample performance on average than other state-of-the-art distributionally robust unit commitment models while staying computationally competent.