Mathematical Sciences: Eigenvalue Optimization and Robust Mathematical Programming

9303772 Burke本项目包括三个研究领域：(1)非线性规划中的鲁棒方法，包括开发涉及不可行性和缺乏规律性问题的算法；(2)非线性互补的最小二乘方法和基于最小二乘方法的全局定义算法的发展；(3)特征值优化和特征值问题。该研究对非线性规划方法的传统应用领域具有重要的拓展意义。例如，不可行的非线性程序的技术可以应用于引导系统通过灾难性事件；例如，由于输电线路下降或开关站损坏，电网中不能再满足需求的潮流维护。非线性互补的全局策略允许求解离散的无限维非线性变分不等式；例如，液晶模拟中的变分不等式。特征值优化变分原理的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. ***