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.
