Stochastic Variational Problems: Optimization and Equilibrium

[20205699]本研究计划是围绕随机规划中的近似问题，特别是在两个相当具有挑战性的问题中出现：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；不只是优化，我们必须找到一种机制来确定一个价格体系，在这个体系下，优化才会发生。这种均衡是依靠不动点定理推导出来的。第三，由于随机优化问题只能求解离散化版本，因此彻底研究近似问题至关重要。不仅是逼近描述不确定性的随机过程的问题，而且是如何从现有数据中改进该过程的构造以及分析这将对随机程序的解产生的影响。随机规划领域为求解和分析不确定性下的决策模型提供了数学工具。本项目将涉及两个重要且困难的应用和近似问题：-地下水修复问题，选择该问题是因为它需要理论和计算的发展。这是一个随机优化问题，系统的状态是通过求解一个系数为快速振荡（非均匀介质）和随机（介质组成的不确定性）的偏微分方程得到的。推导出这个问题的均匀化版本的可能性也将被研究。——不确定环境下的瓦尔拉斯平衡问题。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（建立一个“均衡”），在这个系统下，这种随机优化必须发生。——随机规划中的近似问题。提出了动态随机规划问题参数的可靠估计问题。预期采用一种更全面的方法，利用所有可用的信息，而不仅仅是收集到的数据，将为随机规划问题带来更可靠的解决办法。

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.