Mathematical Sciences: Approximation in Stochastic Programming and Other Variational Problems

9300930 Wets本研究的统一主题是在变分问题中出现的积分泛函的表收敛，特别是随机优化问题，即涉及随机积分泛函的问题。P.I.已经确定了以下将被研究的问题：在加法和表加法下的表收敛的保持，被积不连续的积分函数的表收敛，积分泛函收敛的定量理论的发展。在随机积分泛函的情况下，P.I.将发展随机下半连续函数在有或没有EPI-IIID假设的情况下的表相合性结果，设计可实现的程序来估计近似解的可靠性水平，并利用大偏差理论来获得收敛速度。%优化问题(如稀缺资源分配、最优工程设计、后勤保障行动计划等)如果不求助于近似法，就不能解决。为了验证一个近似方案，人们应该证明，对近似的改进可以产生一个更好的解(收敛问题)，并且只要有可能，就允许计算误差界，以便可以对可能出现的误差(收敛速度)进行估计。P.I.计划调查这样的问题，特别是与所谓的随机优化问题(不确定情况下的模型决策)有关的问题，这类问题很难解决，但对应用来说非常重要。***

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. ***