Developing and Understanding Methods for Nonlinear Optimization

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This project is devoted to research in unconstrained and constrained optimization. The project has five parts: First, tensor methods will be developed for constrained optimization and for large, sparse systems of nonlinear equations. These methods show great promise of yielding general purpose algorithms that are highly efficient on nonsingular and singular problems. Second, trust region methods will be developed for nonlinearly inequality and equality constrained optimization problems that have satisfactory local and global convergence theory even in the presence of linearly dependent constraint gradients, and perform efficiently and robustly in practice. Third, a thorough analysis will be performed of several secant methods for nonlinearly constrained optimization, with the goal of guaranteeing local and superlinear convergence with an arbitrary positive definite initial Hessian approximation. In particular, methods will be investigated that utilize the full Hessian of the Lagrangian, including some new, promising augmented Lagrangian methods that will also be investigated computationally. Fourth, algorithms will be developed to solve the implicit nonlinear least squares problem, an optimization problem that arises in curve fitting or in data fitting when there is no dependent variable. This problem results in a nonlinear equality constrained optimization problem which is expected to be solved by trust region methods. Fifth, some parallel and sequential methods will be investigated for global optimization. These are stochastic algorithms which adaptively partition the feasible region, and lead to improvements in both parallel and sequential computation.