Mathematical Sciences: Approximation in Stochastic Programming and Other Variational Problems

小行星9625787 这个项目的统一主题是收敛，特别是epi收敛，积分函数，因为它们出现在随机优化， 而且在其他变分问题中：最优控制问题， 变分法、数理统计等。的主要原因 集中讨论近似问题的一个重要原因是， 关于积分泛函的性质， 在设计中提出了高效可靠的有限优化例程， 维线性和非线性优化问题，使用这些 解决无限维问题的常规方法受到了阻碍， 我们所知道的关于无限近似的严重缺陷 维数优化问题，特别是有限优化问题 问题 随机优化模型提供了支持决策的工具 在不确定性下。这些都是“决策”问题，其数据只有 在统计学意义上，即，要指定的值存在不确定性 问题的一些参数，最好的情况下， 这些参数的统计信息。 典型随机 优化问题是：选择一个灵活的制造计划 当需求不确定时，资产/负债管理 问题，污染控制的政策制定， 大气条件，结构设计（如桥梁， 摩天大楼）将受到（随机分布的）地震 冲击等。在数学上，随机优化问题非常 很难解决，因为人们必须以某种方式考虑到 这些随机参数的所有可能结果。这就是为什么设计 对于这样的问题，一个近似理论是必不可少的，这就是在 这个项目的核心。

9625787 Wets The unifying theme of this project is the convergence, in particular the epi- convergence, of integral functions as they arise in stochastic optimization, but also in other variational problems: optimal control problems, the calculus of variations, mathematical statistics, etc.. The main reason for concentrating on approximation questions is that although much is known about the properties of integral functionals and much progress has been made in the design of efficient and reliable optimization routines for finite dimensional linear and nonlinear optimization problems, the use of these routines to solve infinite dimensional problems has been hampered by serious shortcomings in what we know about the approximation of infinite dimensional optimization problems, in particular by finite optimization problems. Stochastic optimization models provide tools that support decision making under uncertainty. These are ``decision'' problems whose data is only known in a statistical sense, i.e., there is uncertainty about the values to assign to some of the parameters of the problem, and at best one might have some statistical information about these parameters. Typical stochastic optimization problems are: the choice of a flexible manufacturing plan when the demand is not known with certainty, asset/liability management problems, policy setting for pollution controls that are affected by atmospheric conditions, the design of structures (such as bridges, skyscrapers) that are going to be subjected to (randomly distributed) seismic shocks, etc.. Mathematically, stochastic optimization problems are very difficult to solve because one must, in one way or another, take into account all possible outcomes of these random parameters. That is why the design of an approximation theory for such problems is essential, and that is what is at the core of this project.