Collaborative Research: Uncountable Markov Decision Processes and their Applicatioins 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。该项目开发了分析和优化具有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，本课题还研究了转移速率无界的问题。马尔可夫决策过程是一种基本的随机优化模型，广泛应用于生产和服务系统的控制和管理。本项目将开发的方法将填补文献中留下的重要空白，这些空白对应用程序越来越重要。这些差距就是如何控制具有大状态空间的自然和人造系统，以及系统状态快速变化的可能性。这个项目主要有三个方面的任务。任务1将介绍具有一般状态空间、弱连续转移概率和一般动作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，这为学者和实践者提供了解决此类问题的通用工具。任务2将研究具有无界过渡率的连续时间问题。即使在状态空间是离散的情况下，这个一般问题也已经打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，动作集上没有先验界限的情况变得越来越常见。我们的工作将解决这些问题。第三项任务致力于将我们前两项任务的结果应用到几个领域。我们将研究随机库存控制和收入管理问题。如果成功，该项目将为广泛类别的随机系统开发优化方法，并为解决大规模库存和收入管理问题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.