Stochastic Variational Problems: Optimization and Equilibrium

随机变分问题：优化和均衡

• 批准号: 0205699
• 负责人: Roger Wets
• 金额: $ 16.92万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205699&HistoricalAwards=false
• 关键词: Stochastic; Variational; Problems; Optimization; Equilibrium

0205699WetsThis research proposal is centered around approximation issues in stochastic programming, in particular as they arise in two quite challenging problems: equilibria problems in a stochastic environment and recourse problems involving partial differential equations. The "stochastic" equilibrium problem adds a new level of difficulty; rather than just optimizing one must find a mechanism to determine a price system under which the optimization takes place. Such equilibria have been derived by relying on fixed point theorems. Thirdly, because it is only possible to solve discretized versions of stochastic optimization problems, it is of paramount importance to investigate thoroughly approximation issues. Not only the question of approximating the stochastic process that describes the uncertainty but also how to improve the construction of this process from the available data and to analyze the effect this will have on the solution of the stochastic program.The field of stochastic programming provides mathematical tools for solving and analyzing models for decision making under uncertainty. This project will be concerned with two significant and difficult applications and with approximation issues: -- A problem in groundwater remediation which was selected because it requires both theoretical and computational developments. It is a stochastic optimization problem where the state of the system is obtained by solving a partial differential equation whose coefficients are rapidly oscillating (heterogeneous media) and stochastic (uncertainty about the media composition). The possibility of deriving a homogenized version of this problem will also be investigated. -- Walras equilibrium problem in an uncertain environment. This problem is selected because it adds a dimension to stochastic optimization in that one must also find price systems (setting up an "equilibrium") under which this stochastic optimization must take place. -- Approximation issues in stochastic programming. The question of having a reliable estimate for the parameters of a dynamic stochastic programming problem is raised. It is expected that a more comprehensive, approach which makes use of all the information available, rather than just the collected data will result in more reliable solutions for stochastic programming problems.

0205699 WetsThis研究建议是围绕随机规划中的近似问题，特别是因为它们出现在两个相当具有挑战性的问题：在随机环境中的平衡问题和涉及偏微分方程的追索权问题。 “随机”均衡问题增加了一个新的难度;而不仅仅是优化一个必须找到一个机制来确定一个价格体系下的优化发生。这样的平衡是由不动点定理导出的。 第三，因为它是唯一可能解决离散版本的随机优化问题，它是至关重要的，彻底调查近似问题。不仅是描述不确定性的随机过程的近似问题，而且是如何从可用数据中改进该过程的构造并分析其对随机规划的解的影响的问题。随机规划领域为解决和分析不确定性下的决策模型提供了数学工具。该项目将关注两个重要的和困难的应用和近似问题：-地下水修复的问题，因为它需要理论和计算的发展。这是一个随机优化问题，其中系统的状态是通过求解偏微分方程获得的，该偏微分方程的系数是快速振荡的（非均匀介质）和随机的（介质成分的不确定性）。导出一个均匀化版本的这个问题的可能性也将被调查。 不确定环境下的Walras均衡问题选择这个问题是因为它为随机优化增加了一个维度，因为人们还必须找到价格系统（建立“均衡”），在该价格系统下，这种随机优化必须发生。 --随机规划中的近似问题。提出了对动态随机规划问题的参数进行可靠估计的问题。预计一个更全面的方法，利用所有可用的信息，而不仅仅是收集的数据将导致更可靠的随机规划问题的解决方案。