Research on Stochastic Processes and Optimization

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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.