Data-Driven Distributionally Robust Stochastic Programming

随机规划有助于解决有许多未知因素的难题。它依靠概率分布在数学上表示和预测不确定事件。然而，在现实生活中，可能结果的概率很少为人所知。分布鲁棒优化的目标是得到存在这种分布不确定性的解。形成分布鲁棒随机规划的方法有很多种。然而，对于哪种类型的数据、系统或决策者使用哪种类型的模型还没有得到很好的理解。该奖项支持研究对这一基本问题有更深入的理解，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，包括气候数据、水文模拟、专家意见等。研究结果如果成功，将改善水资源管理，使美国社会和经济受益。研究结果将被纳入随机优化的教材中。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，并吸引女性进入工程领域。为了解决有效建模的问题，该项目将尝试以一种方式对模型进行分类，强调不同的数据来源和问题特征可能需要不同的问题表述。本研究任务将运用概率论、统计学和风险理论提出建议。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揭示的不确定性问题，该项目将研究数据驱动的多阶段分布鲁棒随机程序。这项研究任务将研究如何将多周期不确定性转化为模型，并研究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，该项目将检验数据的价值以及不同场景对最优解和值的影响。最后，该项目将运用建模、算法和理论成果来解决现实世界的多期水资源分配问题。如果成功，研究结果也适用于具有复杂多期不确定性的能源、交通和金融等其他问题。

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.