UC San Diego

The goal of this dissertation is to investigate the formulation and analysis of a trust- region interior-point method for solving nonconvex optimization problems with a mixture of equality and inequality constraints. The proposed method is based on minimizing a merit function that may be interpreted as a shifted primal-dual penalty-barrier function. The method generates a sequence of iterates with limit points that are either infeasible stationary points or complementary approximate Karush-Kuhn-Tucker points, i.e., every limit point satisfies reasonable stopping criteria and is a Karush-Kuhn-Tucker point under a regularity condition that is the weakest constraint qualification associated with sequential optimality conditions. Under suitable additional assumptions, the method is equivalent to a shifted variant of the primal-dual path-following method in the neighborhood of a solution.The proposed method has an inner/outer iteration structure. The outer iteration specifies the form of the merit function. The inner iteration optimizes the merit function with fixed parameters using a trust-region method. The algorithm for solving the trust-region subproblem involves a procedure based on the application of a one-dimensional Newton’s method. Methods are proposed for treating the so-called ”hard case” in which no root of the one dimensional equation exists.