Trust Region Methods for Minimization.

This research project investigated a number of topics in unconstrained optimization, constrained optimization, and solving systems of nonlinear equations. The biggest accomplishment was the development of a new class of of methods, called tensor methods, for solving systems of nonlinear equations. These methods led to large increases in efficiency over standard methods on extensive batteries of test problems, with especially large gains on problems with singular Jacobians at the solution. The other major accomplishment was the development of a unified theory of trust region methods for unconstrained optimization. Our theory shows how line search, dogleg, or optimal step methods can be constructed that satisfy first and second order conditions for convergence. Research was also completed on conic methods for optimization, on secant methods that satisfy multiple secant equations, and on issues concerned with the computation of null space bases in constrained optimization. Research was initiated on computational methods for nonlinear least squares problems with errors in the independent variables, and in parallel algorithms for optimization.