Collaborative Research: Uncountable Markov Decision Processes and their Applications to Optimization of Large-Scale Stochastic Systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这个项目开发了分析和优化一般状态和动作集的马尔可夫决策过程的新方法。在连续时间马尔可夫决策过程的情况下，这个项目也研究了无界转移率的问题。马尔可夫决策过程是一种基本的随机优化模型，广泛应用于各种应用，包括生产和服务系统的控制和管理。 在这个项目中开发的方法将填补文献中留下的重要空白，这些空白对应用越来越重要。这些差距是如何控制自然和人造系统与大的状态空间和系统状态的快速变化的可能性。该项目侧重于三项任务。任务1将涵盖一般状态空间、弱连续转移概率和一般行动集的马尔可夫决策过程。这项任务将发展新的理论概念和方法。此外，由于弱连续转移概率涵盖了大多数库存控制和收入管理问题，这为学者和从业人员提供了解决此类问题的通用工具。任务2将研究具有无界转移率的连续时间问题。 即使在状态空间是离散的情况下，这个一般问题也已经公开了相当长的一段时间。然而，在并行处理和分布式计算领域，动作集没有先验界限变得越来越普遍。我们的工作将解决这些问题。第三个任务致力于将我们的结果从前两个任务应用到几个领域。我们将研究随机库存控制和收益管理问题。 如果成功，该项目将为广泛的随机系统开发优化方法，并为大规模库存和收入管理问题的解决方案提供理论和计算基础。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project develops new methods for analysis and optimization of Markov Decision Processes with general state and action sets. In the case of continuous time Markov Decision Processes, this project also studies problems with unbounded transition rates. A Markov Decision Processes is a fundamental stochastic optimization model broadly used in various applications including control and management of production and service systems. The methods to be developed in this project stand to fill important gaps left in the literature that are becoming increasingly more crucial to applications. These gaps are how to control natural and man-made systems with large state spaces and with the possibility of rapid changes of the system states. This project focuses on three tasks. Task 1 will cover Markov Decision Processes with general state spaces, weakly continuous transition probabilities, and general action sets. This task will develop new theoretical concepts and methods. Moreover, since weakly continuous transition probabilities covers most inventory control and revenue management problems, this provides academics and practitioners with a general tool to solve such problems. Task 2 will study continuous time problems with unbounded transition rates. Even in the case when the state space is discrete, this general problem has been open for quite some time. However, in the areas of parallel processing and distributed computing it is becoming increasingly more common to not have an a priori bound on the action set. Our work will address these issues. The third task is dedicated to applying our results from the first two tasks to several areas. We shall investigate stochastic inventory control and revenue management problems. If successful, this project will develop optimization methods for broad classes of stochastic systems and provide theoretical and computation foundations for solutions of large-scale inventory and revenue management problems.