Abstract : The objective of this project was to derive global convergence theorems and practical numerical algorithms for probability-one homotopies in the area of constrained optimization, large sparse unstructured systems of nonlinear equations, H2 control design and analysis, and H^2/H infinity disturbance and rejection controls. The emphasis was on useful convergence theorems expressed in terms of conditions on the original problem rather than conditions on the derived problem. Important applications were considered as a guide to what theoretical questions should be asked.