Data-Driven Adjustable Robust Optimization

TL;DR

This paper introduces a novel data-driven approach for adjustable robust optimization that adapts uncertainty sets to improve feasibility and solution robustness, applicable to stochastic and non-stochastic uncertainties, demonstrated on power flow problems.

Contribution

It develops a two-stage method to synthesize and adjust uncertainty sets based on data, extending robust optimization to distributionally robust scenarios with finite reformulations.

Findings

Enhanced feasible solution space for uncertain problems

Effective handling of stochastic and non-stochastic uncertainties

Successful application to power flow optimization

Abstract

In this paper, we develop a two-stage data-driven approach to address the adjustable robust optimization problem, where the uncertainty set is adjustable to manage infeasibility caused by significant or poorly quantified uncertainties. In the first stage, we synthesize an uncertainty set to ensure the feasibility of the problem as much as possible using the collected uncertainty samples. In the second stage, we find the optimal solution while ensuring that the constraints are satisfied under the new uncertainty set. This approach enlarges the feasible state set, at the expense of the risk of possible constraint violation. We analyze two scenarios: one where the uncertainty is non-stochastic, and another where the uncertainty is stochastic but with unknown probability distribution, leading to a distributionally robust optimization problem. In the first case, we scale the uncertainty set and find the best subset that fits the uncertainty samples. In the second case, we employ the Wasserstein metric to quantify uncertainty based on training data, and for polytope uncertainty sets, we further provide a finite program reformulation of the problem. The effectiveness of the proposed methods is demonstrated through an optimal power flow problem.