Stochastic Variational Problems: Equilibrium & Modeling Uncertainty

中文（简体） 研究者研究了两类与随机优化问题相关的问题--不确定性下的决策问题。 两者都带来了非平凡的建模，理论和计算挑战。 第一部分的项目涉及扩展的随机规划和变分分析方法，以处理一大类变分问题的随机环境中，这可能被称为随机平衡问题。 它们包括变分不等式，合作和非合作游戏，Walras型平衡，解决包含和不动点问题，当一些组件的问题是随机的。 这里的目标是发展一个理论，导致可实施的numericalprocedures。 该项目的第二部分涉及使themethodology的随机规划更容易获得更广泛的用户，即与程序的设计，允许用户描述的不确定性时，只有（非常）有限的数据是关于随机参数。 这是特别重要的不确定性是揭示超时（或空间）和决策者可以使用此信息来选择追索权（调整）的决定。 几乎所有重要的决策问题，除了可能是一些纯粹的技术性质，涉及到一些参数或组件的问题的不确定性水平。 例如，（a）制定货币或经济政策需要考虑经济和金融市场的不确定性，（B）选择制造业的主调度计划需要考虑市场反应的不确定性，（c）选择环境保护计划需要平衡不断变化的大气条件与社会和经济影响，所有这些都涉及各种不确定性，（d）与能源供应有关的政策问题需要考虑到世界市场价格的变幻莫测以及潜在的技术创新等等。这些问题的确定性模型，即使包括情景“分析”，也会导致所谓的最优决策，而这些决策几乎从来都不是稳健的，有时甚至是严重误导的。 因此，在建立优化模型或平衡模型时，考虑不确定性是至关重要的。 使用变分分析工具，算法程序开发解决（非凸，有限维）变分不等式，和近似计划随机优化问题，调查员开发的理论结果和计算工具，以解决suchfundamental问题，并提高建模的不确定性组件。 一个重要的问题，无论是随机优化和随机环境中的平衡问题，是不确定性的建模。 在实际情况下，并不总是有足够的数据来证明使用经典的统计方法是正确的。 分析器研究“缺乏数据”的问题，检查允许用户在估计未知参数的分布时结合其他信息的技术。

Wets0705470 The investigator studies two classes of issues related tostochastic optimization problems -- decision-making underuncertainty. Both come with nontrivial modeling, theoretical,and computational challenges. The first part of the projectdeals with an extension of the stochastic programming andvariational analysis methodology to deal with a large class ofvariational problems in a stochastic environment, which could becalled stochastic equilibrium problems. They include variationalinequalities, cooperative and noncooperative games, Walras-typeequilibria, solving inclusions, and fixed point problems, whensome of the components of the problem are stochastic. The goalhere is to develop a theory that leads to implementable numericalprocedures. The second part of the project deals with making themethodology of stochastic programming more accessible to a widerrange of users, namely with the design of procedures that allowthe user to describe the uncertainty when only (very) limiteddata are available about the random parameters. This isparticularly critical when the uncertainty is to be revealed overtime (or space) and the decision maker can use this informationto select recourse (adjusting) decisions. Almost all important decision-making problems, exceptpossibly a few that are purely technical in nature, involve somelevel of uncertainty about some of the parameters or componentsof the problem. For example, (a) setting monetary or economic policy requires taking into account uncertainty in the economic and financial markets, (b) selecting a master scheduling plan in manufacturing requires taking into account uncertainty in the market's response, (c) the choice of an environmental protection plan needs to balance changing atmospheric conditions with sociological and economic impacts, all involving various uncertainties, (d) policies issues related to energy supply require taking into account world market price vagaries as well as potential technological innovations,and so on. Deterministic models for such problems, even whenincluding scenario "analysis," lead to so-called optimaldecisions that almost never are robust, sometimes are evenseriously misleading. So it is crucial to take uncertainty intoaccount when setting up either optimization or equilibriummodels. Using tools of variational analysis, algorithmicprocedures developed for solving (nonconvex, finite-dimensional)variational inequalities, and approximation schemes forstochastic optimization problems, the investigator developstheoretical results and computational tools to solve suchfundamental problems and to improve the modeling of theuncertainty components. An important issue, both for stochasticoptimization and for equilibrium problems in a stochasticenvironment, is the modeling of the uncertainty. In practicalsituations, sufficient data are not always available to justifythe use of classical statistical methods. The investigatorstudies "lack of data" issues, examining techniques that allowthe user to incorporate other information in the estimation ofthe distribution of the unknown parameters.