Data-Driven Distributionally Robust Stochastic Programming

随机规划有助于解决具有许多未知因素的难题。它通过依赖概率分布在数学上表示和预测不确定事件来做到这一点。然而，在现实生活中，可能结果的概率很少为人所知。分布稳健优化的目标是在存在这种分布不确定性的情况下获得解。有多种方法可以形成分布健壮的随机程序。然而，哪种类型的模型用于哪种类型的数据、系统或决策者并不是很好的理解。该奖项支持研究对这一根本问题有更深的理解，并探索多时期的不确定性。该项目考虑了长期水资源管理问题，这些问题需要各种来源的投入，包括气候数据、水文模拟、专家意见等。如果结果成功，将改善水资源管理，使美国社会和经济受益。研究成果将被纳入关于随机优化的教材中。因此，该项目将有助于教育学生。水的应用将被用来展示我们领域的社会影响，并吸引女性从事工程。为了解决有效建模的问题，该项目将尝试对模型进行分类，以突出不同的数据来源和问题特征可能需要不同的问题公式。本研究课题将运用概率论、统计学、风险学等理论提出建议。然后，它将利用这些结果来改进建模和数据收集，并设计抽样方案。为了解决随着时间的推移而暴露出来的不确定性问题，该项目将研究数据驱动的多阶段分布式稳健随机程序。这项研究任务将研究如何将多周期不确定性转化为模型，并调查由此产生的模型结构和性质。为了有效地解决这些模型，将探索基于分解的解决方法。此外，该项目将审查数据的价值以及不同情景对最佳解决方案和价值的影响。最后，该项目将实施建模、算法和理论研究成果，以解决现实世界中的多时段水资源分配问题。如果成功，该结果也适用于能源、交通、金融等具有复杂多时期不确定性的其他问题。

Stochastic programming aids in solving difficult problems with many unknown factors. It does so by relying on probability distributions to mathematically represent and predict uncertain events. However, probabilities of possible outcomes are rarely known in real life. Distributionally robust optimization aims to obtain solutions in the presence of such distributional uncertainties. There are a variety of ways to form distributionally robust stochastic programs. However, which type of model to use for which type of data, system, or decision maker is not well understood. This award supports research to have a deeper understanding of this fundamental question and to explore multi-period uncertainties. The project considers long-term water resources management problems that take various sources of input including climate data, hydrological simulations, expert opinions, and so forth. The results, if successful, will yield improved water management, benefitting the U.S. society and economy. The research findings will be incorporated into educational materials on stochastic optimization. The project will therefore contribute to educating students. The water application will be used to demonstrate the societal impact of our field and to attract women to engineering.To address the problem of effective modeling, the project will attempt a classification of models in a way that highlights how different sources of data and problem characteristics may require differing problem formulations. This research task will use probability theory, statistics, and risk theory to make recommendations. It will then utilize these results to improve modeling and data collection and devise sampling schemes. To address the problem of uncertainty revealed over time, the project will investigate data-driven multistage distributionally robust stochastic programs. This research task will examine how to translate multi-period uncertainties into the model and investigate the resulting model structure and properties. To effectively solve these models, decomposition-based solution methodologies will be explored. In addition, the project will examine the value of data and the effect of different scenarios on optimal solutions and values. Finally, the project will implement the modeling, algorithmic, and theoretical findings to solve real-world multi-period water allocation problems. If successful, the results are also applicable to other problems in energy, transportation, and finance with complex multi-period uncertainties.