Research on Stochastic Processes and Optimization

随机过程与优化研究

• 批准号: 9704426
• 负责人: Paul Dupuis
• 金额: $ 12.43万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704426&HistoricalAwards=false
• 关键词: Research; Stochastic; Processes; Optimization

9704426 Dupuis This project covers three topics: (i) theory and applications of the Skorokhod Problem; (ii) large deviations and risk sensitive control and robust control of queueing networks; and (iii) computational methods for deterministic optimal control problems and related first order nonlinear PDE. The solution to the Skorokhod Problem defines what might be considered the input/output map for a number of exact and approximate models in queueing, communication, economics, and other areas. Except in special cases, little is known about the analytical properties of this mapping. This research will build on prior work of the investigator and H. Ishii which takes a geometric approach to obtaining regularity conditions for the Skorokhod Problem. The main thrust of the present work is the development and application of methods based on convex duality. The second part of the project studies large deviations for queueing networks. The main topic here is the formulation and analysis of risk sensitive criteria for the control and regulation of queueing networks. In almost all models for networks there are errors, approximations, and model uncertainty, and one would like to design control schemes that are insensitive to such errors. Using the duality between exponential integrals and the relative entropy function, one can (at least in the context of diffusions) give a precise quantitative characterization of the additional robustness properties that are obtained when a risk sensitive criteria is used. The investigator will formulate and analyze via large deviation techniques such criteria in a queueing context. The final part of the project considers cmmputational methods for deterministic optimal control problems. The main emphasis here is on the development of practical algorithms with good qualitative properties. This will be done for a class of problems with very similar features that arise in large deviation for diffusions, certain problems fr om computer vision, and robust nonlinear control and robust filtering. Modern communication, computer, and queueing systems are very complicated, and in fact too complicated to analyze in complete detail. As systems become more and more heterogeneous (e.g., different data classes, different quality of service requirements, etc.) the control and regulation of these systems becomes more difficult and less intuitive. The first two parts of this project investigate two approaches to dealing with such systems. When analyzing queueing and communication systems one needs (relatively) simple system models which capture the most important aspects of the true system. Linear systems are unable to capture the system behavior caused by hard constraints (non-negativity of queue lengths, limits on buffer sizes, etc.) and discontinuities that occur naturally in well designed routing and service protocols. Part of this research is to look at alternatives to linear systems. The second part focuses on the study of risk sensitive criteria for such systems. Risk sensitive and robust criteria provide alternatives to traditional criteria for the evaluation of system performance, and it is now understood that they are very useful in situations where model uncertainty and robustness against modeling errors are important. Such robustness properties are always important for communication and manufacturing systems. The aim here is to properly formulate and analyze risk-sensitive criteria for random networks, and to quantify the robustness properties when compared to traditional criteria. The last part of the project is devoted to the development of usable and efficient computational methods for deterministic optimal control problems, and in particular to the class of problems that arise in the second part of the proposal.

9704426 DUPUI这个项目涵盖了三个主题：（i）Skorokhod问题的理论和应用； （ii）大偏差和风险敏感控制和对排队网络​​的强大控制； （iii）确定性最佳控制问题和相关一阶非线性PDE的计算方法。 Skorokhod问题的解决方案定义了可以被认为是排队，交流，经济学和其他领域的许多精确模型的输入/输出图。除在特殊情况下，对此映射的分析特性知之甚少。这项研究将基于研究人员和H. ishii的先前工作，该工作采用几何方法来获得Skorokhod问题的规律性条件。本工作的主要目的是基于凸双重性的方法的开发和应用。 项目的第二部分研究排队网络的大偏差。这里的主要主题是对排队网络​​控制和调节的风险敏感标准的制定和分析。在几乎所有网络模型中，都有错误，近似值和模型不确定性，并且希望设计对此类错误不敏感的控制方案。使用指数积分和相对熵函数之间的二元性，一个人可以（至少在扩散的背景下）对使用风险敏感标准时获得的附加鲁棒性特性进行精确的定量表征。 研究人员将在排队的情况下通过大偏差技术制定和分析。该项目的最后一部分考虑了确定性最佳控制问题的CMMPutational方法。这里的主要重点是具有良好定性特性的实用算法的发展。这将用于一类具有非常相似特征的问题，这些特征在扩散较大的偏差，计算机视觉以及强大的非线性控制和稳健的过滤中出现。 现代通信，计算机和排队系统非常复杂，实际上太复杂了，无法详细分析。随着系统变得越来越异质（例如，不同的数据类别，不同的服务要求等不同），对这些系统的控制和调节变得更加困难，更直观。该项目的前两个部分研究了处理此类系统的两种方法。 在分析排队和通信系统时，需要（相对）简单的系统模型，这些系统模型捕获了真实系统的最重要方面。线性系统无法捕获由硬约束（队列长度的非负性，缓冲区尺寸的限制等）引起的系统行为，并且在设计良好的路由和服务协议中自然发生的不连续性。这项研究的一部分是研究线性系统的替代方案。第二部分重点是研究此类系统的风险敏感标准。 风险敏感和健壮的标准为评估系统性能的传统标准提供了替代方案，现在可以理解，在模型不确定性和鲁棒性抗建模错误的情况下它们非常有用。 这种鲁棒性特性对于通信和制造系统始终很重要。这里的目的是正确制定和分析随机网络的风险敏感标准，并与传统标准相比量化鲁棒性特性。 该项目的最后一部分致力于开发确定性最佳控制问题的可用和有效的计算方法，尤其是提案第二部分中出现的问题类别。