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

Watson 9626968研究人员利用高质量的数学软件为约束优化、大型稀疏非结构非线性方程组、多项式方程组和各种干扰抑制控制领域的概率1同伦推导了全局收敛定理并开发了实用的数值算法。重点是用关于原始问题的条件而不是关于派生问题的条件来表示有用的收敛定理。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。该数学软件使用HOMPACK中现有的曲线跟踪内核，利用HOMPACK构建特定于应用程序的外壳。物理现象的真实模型通常都很大，而且是非线性的。求解由这种模型产生的大型非线性方程组需要新的算法，最近发展起来的一些同伦算法已经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。这项工作研究了这些新的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。

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.