Mathematical Sciences: Approximation in Stochastic Programming and Other Variational Problems

数学科学：随机规划和其他变分问题中的近似

• 批准号: 9300930
• 负责人: Roger Wets
• 金额: $ 9.6万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9300930&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Approximation; Stochastic; Programming

9300930 Wets The unifying theme of this research is the epi-convergence of integral functionals as they arise in variational problems, in particular stochastic optimization, i.e., problems involving random integral functionals. The P.I. has identified the following questions that will be investigated: the preservation of epi-convergence under addition and epi-addition, the epi- convergence of integral functions with discontinuous integrands, the development of a quantitative theory for the convergence of integral functionals. In the case of random integral functionals, the P.I. will develop epi-consistency results for random lower semicontinuous functions with and without the epi- iid assumption, design implementable procedures to estimate the reliability level of approximate solutions and exploit the theory of large deviations to obtain convergence rates. %%% Optimization problems (such as the allocation of scare resources, optimal engineering design, planning of logistic support operations, etc.) cannot be solved without resorting to approximations. To validate an approximation scheme one should demonstrate that a refinement of the approximation yields a better solution (convergence questions) and, whenever possible, allows for the calculation of error bounds so that one can have an estimate of the error that may occur (convergence rate). The P.I. plans to investigate such questions, in particular in connection with so-called stochastic optimization problems (that model decision making under uncertainty) a class of problems that is very difficult to solve, but are very important as far as applications are concerned. ***

小行星9300930 这项研究的统一主题是积分泛函的epi-convergence，因为它们出现在变分问题中，特别是随机优化，即，涉及随机积分泛函的问题。 私家侦探已经确定了以下问题，将进行调查：epi收敛下的加法和epi加法，epi收敛的积分函数与不连续的被积函数，发展的定量理论的收敛积分泛函。 在随机积分泛函的情况下，P.I.将开发epi-consistency结果随机下连续函数和没有epi-iid假设，设计可实现的程序来估计 近似解的可靠性水平，并利用大偏差理论获得收敛速度。 %%% 优化问题（如稀缺资源的分配、最优工程设计、后勤保障行动计划等）不采用近似法是无法解决的。 为了验证近似方案，应该证明近似的细化产生更好的解决方案（收敛问题），并在可能的情况下，允许计算误差范围，以便可以估计可能发生的误差（收敛速度）。 私家侦探计划研究这些问题，特别是与所谓的随机优化问题（在不确定性下进行决策的模型）有关的问题，这类问题很难解决，但就应用而言非常重要。 ***