Research on Stochastic Processes and Optimization

9704426 Dupuis本项目包括三个主题：(i) Skorokhod问题的理论与应用；（ii）排队网络的大偏差、风险敏感控制和鲁棒控制；（iii）确定性最优控制问题和相关一阶非线性偏微分方程的计算方法。Skorokhod问题的解决方案定义了在排队、通信、经济和其他领域的许多精确和近似模型的输入/输出映射。除特殊情况外，人们对这种映射的解析性质所知甚少。本研究将以研究者和H. Ishii先前的工作为基础，采用几何方法来获得Skorokhod问题的正则性条件。目前工作的主要推力是基于凸对偶的方法的发展和应用。项目的第二部分研究排队网络的大偏差。本文的主要课题是制定和分析排队网络控制和调节的风险敏感准则。在几乎所有的网络模型中都存在误差、近似和模型不确定性，人们希望设计对这些误差不敏感的控制方案。利用指数积分和相对熵函数之间的对偶性，人们可以（至少在扩散的背景下）给出当使用风险敏感标准时获得的附加鲁棒性的精确定量表征。研究者将通过大偏差技术在排队环境中制定和分析这样的标准。项目的最后一部分考虑了确定性最优控制问题的计算方法。这里的重点是开发具有良好定性性质的实用算法。这将用于一类具有非常相似特征的问题，这些问题在扩散的大偏差中出现，某些来自计算机视觉的问题，以及鲁棒非线性控制和鲁棒滤波。现代通信、计算机和排队系统非常复杂，事实上，它们太复杂了，无法进行详细的分析。随着系统变得越来越异构（例如，不同的数据类、不同的服务质量需求等），这些系统的控制和调节变得越来越困难，也越来越不直观。本项目的前两部分研究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）简单的系统模型来捕捉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、缓冲区大小的限制等）和在设计良好的路由和服务协议中自然发生的不连续性引起的系统行为。这项研究的一部分是寻找线性系统的替代方案。第二部分重点研究了此类系统的风险敏感准则。风险敏感和健壮的标准为系统性能的评估提供了传统标准的替代方案，并且现在可以理解，它们在模型不确定性和对建模错误的健壮性很重要的情况下非常有用。这种鲁棒性对于通信和制造系统总是很重要的。这里的目的是适当地制定和分析随机网络的风险敏感标准，并量化与传统标准相比的鲁棒性。项目的最后一部分致力于为确定性最优控制问题开发可用且有效的计算方法，特别是针对提案第二部分中出现的一类问题。

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.