Harmonizing SAA and DRO

TL;DR

This paper introduces a novel harmonizing optimization (HO) method that adaptively combines SAA and DRO to improve decision-making under uncertainty across different data sizes, with proven guarantees and practical benefits.

Contribution

The paper proposes a new HO approach that dynamically balances SAA and DRO, providing finite-sample guarantees and asymptotic optimality without classifying data size.

Findings

HO outperforms Wasserstein-based DRO in solution quality.

HO effectively improves scenario reduction and approximation accuracy.

HO adapts seamlessly to varying data sizes without prior classification.

Abstract

Decision-makers often encounter uncertainty, and the distribution of uncertain parameters plays a crucial role in making reliable decisions. However, complete information is rarely available. The sample average approximation (SAA) approach utilizes historical data to address this, but struggles with insufficient data. Conversely, moment-based distributionally robust optimization (DRO) effectively employs partial distributional information but can yield conservative solutions even with ample data. To bridge these approaches, we propose a novel method called harmonizing optimization (HO), which integrates SAA and DRO by adaptively adjusting the weights of data and information based on sample size N. This allows HO to amplify data effects in large samples while emphasizing information in smaller ones. More importantly, HO performs well across varying data sizes without needing to classify them as large or small. We provide practical methods for determining these weights and demonstrate that HO offers finite-sample performance guarantees, proving asymptotic optimality when the weight of information follows a 1/\sqrt{N}-rate. In addition, HO can be applied to enhance scenario reduction, improving approximation quality and reducing completion time by retaining critical information from reduced scenarios. Numerical results show significant advantages of HO in solution quality compared to Wasserstein-based DRO, and highlight its effectiveness in scenario reduction.