Collaborative Research: Semi-Infinite Probabilistic Optimization

该研究旨在完善高不确定性下的非线性随机优化的数学基础。我们计划开发新的模型，涉及无限多个函数的约束，以刻画随机结果的分布。特别地，我们将考虑带有概率分布函数及其变换的约束的半无限优化问题，例如过剩/短缺度量。我们将分析这些模型的结构，并发展最优性条件和对偶理论。此外，我们还将对这些问题进行稳定性分析，并开发新的逼近方法。这些理论进展将为发展新的数值方法来解决高不确定性下的优化问题奠定基础。在许多实际问题中，必须在不确定条件下做出最优决策。投资计划就是一个例子，但这类问题出现在电信、保险和金融、发电和配电、供应链管理、制造业、军事以及其他领域。当存在不太可能发生但不能安全地忽略的事件时，现有技术不能很好地工作。该项目将提供新的数学工具和数值方法来处理这类问题。

The proposed research aims at improving the mathematical foundations of nonlinear stochastic optimization under high uncertainty. We plan to develop new models involving infinitely many constraints on functions characterizing distributions of random outcomes. In particular we will consider semi-infinite optimization problems with constraints on the probability distribution functions and their transforms, e.g., excess/shortfall measures. We will analyze the structure of these models and we will develop optimality conditions and duality theory. Further we will carry out an analysis of stability and develop new approximation methods for these problems. These theoretical advances will serve as a basis for developing new numerical methods for solving optimization problems under high uncertainty.In many practical problems optimal decisions must be made under uncertain conditions. Investment planning is one example, but problems of this type occur in telecommunications, insurance and finance, electricity generation and distribution, supply chain management, manufacturing, and in the military, as well as in other areas. Existing techniques do not work well when there are events that are very unlikely to occur but which cannot be safely ignored. This project will provide new mathematical tools and numerical methods to deal with such problems.