Mathematical Sciences: Theory and Application of Homotopy Techniques in Nonlinear Programming

数学科学：非线性规划同伦技术的理论与应用

• 批准号: 9625968
• 负责人: Layne Watson
• 金额: $ 8万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625968&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Theory; Application; Homotopy

Watson 9626968 The investigator derives global convergence theorems and develops practical numerical algorithms with high quality mathematical software for probability-one homotopies in the areas of constrained optimization, large sparse unstructured systems of nonlinear equations, polynomial systems of equations, and various disturbance rejection controls. The emphasis is on useful convergence theorems expressed in terms of conditions on the original problem rather than conditions on the derived problem. Important applications in control systems design and multidisciplinary design optimization are used to guide the theoretical investigations. The mathematical software uses the existing curve tracking kernels in HOMPACK, leveraging HOMPACK to build application-specific shells. Realistic models of physical phenomena are typically large and nonlinear. Solving the large nonlinear systems of equations resulting from such models requires new algorithms, and some recently developed homotopy algorithms have been successfully applied to problems in circuit simulation, aircraft design optimization, composite material design, control system synthesis, and manufacturing mechanism design. This work studies the theoretical properties of these new homotopy algorithms, and develops practical, high quality mathematical software that provides a useful tool for engineers and scientists.

沃森9626968研究人员推导出全局收敛定理，并开发实用的数值算法与高品质的数学软件的概率一同伦在约束优化，大型稀疏非结构化系统的非线性方程，多项式系统的方程，和各种干扰抑制控制。 重点是有用的收敛定理表示的条件对原问题，而不是条件的衍生问题。 在控制系统设计和多学科设计优化中的重要应用被用来指导理论研究。 数学软件使用HOMPACK中现有的曲线跟踪内核，利用HOMPACK构建特定于应用程序的shell。 物理现象的现实模型通常是大的和非线性的。 求解由这种模型产生的大型非线性方程组需要新的算法，最近开发的同伦算法已成功地应用于电路仿真、飞机设计优化、复合材料设计、控制系统综合和制造机构设计等问题。 本文研究了这些新同伦算法的理论性质，并开发了实用的、高质量的数学软件，为工程师和科学家提供了有用的工具。