Stochastic Variational Problems: Equilibrium & Modeling Uncertainty

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。两者都带来了重要的建模、理论和计算挑战。项目的第一部分涉及随机规划和变分分析方法的扩展，以处理随机环境中的一大类变分问题，这些问题可称为随机平衡问题。它们包括变分不等式、合作和非合作博弈、瓦尔拉斯型均衡、求解包含和不动点问题（当问题的某些组成部分是随机的时候）。这里的目标是发展一种理论，导致可实施的数值程序。该项目的第二部分涉及使更广泛的用户更容易使用随机规划方法，即设计允许用户在只有（非常）有限的随机参数数据可用时描述不确定性的程序。当不确定性随着时间的推移（或空间）被揭示出来，决策者可以使用这些信息来选择追索权（调整）决策时，这一点尤为重要。几乎所有重要的决策问题，除了少数纯粹技术性的问题外，都涉及到问题的某些参数或组成部分的某种程度的不确定性。例如，(a)制定货币或经济政策需要考虑到经济和金融市场的不确定性，(b)选择制造业的总调度计划需要考虑到市场反应的不确定性，(c)环境保护计划的选择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化以及可能的技术革新等等。这些问题的确定性模型，即使包括场景“分析”，也会导致所谓的最优决策，这些决策几乎从来都不是稳健的，有时甚至会严重误导。因此，在建立优化模型或均衡模型时，考虑不确定性是至关重要的。使用变分分析工具，为解决（非凸，有限维）变分不等式开发的算法程序，以及随机优化问题的近似方案，研究者开发理论结果和计算工具来解决这些基本问题，并改进不确定性成分的建模。对于随机优化和随机环境中的平衡问题，一个重要的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经典统计方法的使用是正确的。研究者研究“缺乏数据”问题，检查允许用户在估计未知参数分布时纳入其他信息的技术。

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.