Multistage Stochastic Integer Programming: Approximate Solution Methods and Applications

多阶段随机规划(MSP)为不确定条件下的序贯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，不同资产的收益在投资时具有高度的不确定性)。传统上很难高精度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，我们可以准确地对不确定性进行建模。利用现有数据的能力使将不确定性纳入数学模型成为可能，随机编程就是这种情况。此外，在许多应用中，计划范围具有多个决策阶段，并且不确定性随着时间的推移而逐渐显现。因此，MSP是一种可行的建模方法。*MSP在能源、金融、调度等领域有着广泛的应用。然而，众所周知，MSP模型在一般情况下很难解决，而且现有的解决方法经常无法解决实际大小问题。基于其应用潜力和最新解决方法的局限性，该计划旨在为MSP(特别是整数变量)做出基础性的算法和理论贡献，并将其应用扩展到各种领域。*该计划的主题1将专注于开发能够克服MSP类问题，特别是涉及整数变量的MSP问题的建模和算法挑战的方法，并提供(可证明的)良好的可行策略。该方法将主要基于使用(线性)决策规则的新方法。特别是，将为具有整数变量的MSP模型开发新的决策规则。将分析所提出的方法的可操作性和所获得的解的质量。研究结果将极大地推动随机规划的发展。*节目的主题2将探索MSP的各种应用，如操作室调度、电力系统和投资组合优化。对于这些领域中的某些重要问题，将提出新的MSP模型，并将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规划、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.