标题： 协调SAA和DRO 显示英文标题

标题： Harmonizing SAA and DRO

摘要： 决策者经常遇到不确定性，不确定参数的分布在做出可靠决策中起着关键作用。然而，通常无法获得完整的信息。样本平均近似（SAA）方法利用历史数据来解决这个问题，但在数据不足时表现不佳。相反，基于矩的分布鲁棒优化（DRO）有效地利用了部分分布信息，但即使在数据充足的情况下也可能产生保守的解决方案。为了弥合这些方法，我们提出了一种称为协调优化（HO）的新方法，该方法通过根据样本量N自适应调整数据和信息的权重来整合SAA和DRO。这使得HO在大样本中能够放大数据的影响，而在小样本中则强调信息的重要性。更重要的是，HO在不同数据规模下表现良好，而无需将它们分类为大或小。我们提供了确定这些权重的实用方法，并证明HO提供了有限样本性能保证，当信息权重遵循1/\sqrt{N}的速率时，证明了渐近最优性。此外，HO可以用于增强场景缩减，通过保留缩减场景中的关键信息来提高近似质量和减少完成时间。数值结果表明，与基于Wasserstein的DRO相比，HO在解的质量方面具有显著优势，并突显了其在场景缩减中的有效性。 显示英文摘要

摘要： 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.