Multistage Stochastic Integer Programming: Approximate Solution Methods and Applications

多阶段随机规划（MSP）为不确定条件下的顺序决策提供了一个建模框架。数学规划的大多数应用都假定数据是确定性的。然而，现实世界的问题几乎总是包含一些不确定的参数（例如，在投资组合优化问题中，不同资产的回报在投资时是高度不确定的）。传统上很难准确地预测这种不确定性，但现在随着大量历史记录的存在和数据分析的进步，我们可以准确地模拟不确定性。利用现有数据的能力使得将不确定性纳入数学模型成为可能，这就是随机规划的情况。此外，在许多应用中，规划视野具有多个决策阶段，不确定性随着时间的推移逐渐显现。因此，MSP是一种可行的建模方法。MSP在能源、金融和调度等领域有许多应用。然而，一般来说，MSP模型是出了名的难以解决的，现有的解决方法经常不能解决现实生活中的尺寸问题。鉴于其应用潜力和现有解决方法的局限性，本项目旨在对MSP（特别是整数变量）做出基本的算法和理论贡献，并扩展其在各个领域的应用。该计划的主题1将侧重于开发能够克服MSP问题类中的建模和算法挑战的方法，特别是涉及整数变量的方法，并且可以提供（可证明的）好的可行策略。该方法将主要基于使用（线性）决策规则的新方法。特别地，对于具有整数变量的MSP模型，将开发新的决策规则。本文将分析所提出方法的可追溯性以及所得到的解的质量。研究结果将极大地推动随机规划的发展。项目的主题2将探讨MSP的多种应用，如手术室调度、电力系统和投资组合优化。针对这些领域的一些重要问题，我们将提出新的MSP模型，并研究这些模型相对于确定性和两阶段随机规划模型的价值。研究结果将为决策者提供有价值的计划、调度和操作工具。

Multistage stochastic programming (MSP) provides a modeling framework for sequential decision making under uncertainty. The majority of the application of mathematical programming assumes deterministic data. However, real world problems almost always include some uncertain parameters (e.g., in a portfolio optimization problem, the returns of different assets are highly uncertain at the time of investment). It has been traditionally difficult to predict such uncertainties with a high accuracy, but now with the existence of substantial historical records and advances in data analytics, we can accurately model uncertainty. The ability to exploit available data made it possible to incorporate uncertainty into mathematical models, which is the case in stochastic programming. Moreover, in many applications, the planning horizon has multiple decision stages and the uncertainty is revealed gradually over time. Therefore, MSP is a viable modeling approach. MSP has numerous applications in areas like energy, finance, and scheduling. However, MSP models are notoriously hard to solve in general, and existing solution approaches frequently fail to solve real-life size problems. Motivated by its application potential and limitations of the state-of-the-art solution methods, this program aims to make fundamental algorithmic and theoretical contributions to MSP (especially with integer variables), and extend its applications in a variety of areas.Theme 1 of the program will focus on developing methods that can overcome modeling and algorithmic challenges in the class of MSP problems, especially the ones involving integer variables, and that can provide (provably) good feasible policies. The methodology will be mostly based on novel ways of using (linear) decision rules. In particular, new decision rules will be developed for MSP models with integer variables. The tractability of the proposed methods and the quality of the obtained solutions will be analyzed. The results will significantly advance the state-of-the-art in stochastic programming.Theme 2 of the program will explore diverse applications of MSP such as operating room scheduling, power systems and portfolio optimization. Novel MSP models will be proposed for certain important problems in these areas, and the value of such models over deterministic and two-stage stochastic programming models will be investigated. The results will provide valuable planning, scheduling and operational tools for decision makers.