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

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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.