Trust-region interior-point algorithms for a class of nonlinear programming problems
Vicente, Luis Nunes
Dennis, John E., Jr.
Doctor of Philosophy
This thesis introduces and analyzes a family of trust-region interior-point (TRIP) reduced sequential quadratic programming (SQP) algorithms for the solution of minimization problems with nonlinear equality constraints and simple bounds on some of the variables. These nonlinear programming problems appear in applications in control, design, parameter identification, and inversion. In particular they often arise in the discretization of optimal control problems. The TRIP reduced SQP algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state and adjoint equations. These algorithms result from a successful combination of a reduced SQP algorithm, a trust-region globalization, and a primal-dual affine scaling interior-point method. The TRIP reduced SQP algorithms have very strong theoretical properties. It is shown in this thesis that they converge globally to points satisfying first and second order necessary optimality conditions, and in a neighborhood of a local minimizer the rate of convergence is quadratic. Our algorithms and convergence results reduce to those of Coleman and Li for box-constrained optimization. An inexact analysis is presented to provide a practical way of controlling residuals of linear systems and directional derivatives. Complementing this theory, numerical experiments for two nonlinear optimal control problems are included showing the robustness and effectiveness of these algorithms. Another topic of this dissertation is a specialized analysis of these algorithms for equality-constrained optimization problems. The important feature of the way this family of algorithms specializes for these problems is that they do not require the computation of normal components for the step and an orthogonal basis for the null space of the Jacobian of the equality constraints. An extension of More and Sorensen's result for unconstrained optimization is presented, showing global convergence for these algorithms to a point satisfying the second-order necessary optimality conditions.
Mathematics; Chemical engineering; Operations research