Mathematical Sciences: Eigenvalue Optimization and Robust Mathematical Programming

9303772 Burke这个项目包括三个研究领域：(1)非线性规划的稳健方法，包括为涉及不可行和缺乏正则性的问题开发算法；(2)非线性互补的最小二乘方法，以及基于最小二乘方法的全局定义算法的开发；以及(3)特征值优化和特征值问题。这一研究可以极大地拓展非线性规划方法的传统应用领域。例如，可以应用用于不可行的非线性规划的技术来引导系统通过灾难性事件；例如，维持由于输电线路故障或开关站损坏而不再满足需求的电网中的潮流。非线性互补中的全局策略可能允许解离散的无限维非线性变分不等式；例如，在液晶模拟中发现的变分不等式。特征值优化中的变分原理研究在具有非线性动力学的稳定结构的优化设计中有着广泛的应用。***

9303772 Burke This project comprises three areas of study: (1) robust methods in nonlinear programming, including development of algorithms for problems involving infeasibility and lack of regularity; (2) least-squares methods for nonlinear complementarity, and development of globally defined algorithms based on a least-squares methodology, and (3) eigenvalue optimization and eigenvalue problems. This research can significantly expand the traditional domain of applications for nonlinear programming methods. For example, techniques for infeasible nonlinear programs can be applied to guide a system through a catastrophic event; e.g., the maintenance of power flow in a power grid that can no longer meet demand due to a downed transmission line or a damaged switching station. Global strategies in nonlinear complementarity may permit the solution of discrete infinite dimensional nonlinear variational inequalities; e.g., the variational inequalities found in liquid crystal simulation. The research into variational principles in eigenvalue optimization has wide application in the optimal design of stable structures having nonlinear dynamics. ***