UC San Diego

This dissertation focuses on cubic regularization methods for the globalization of Newton's method, with applications in unconstrained and constrained optimization. In recent years, cubic regularization algorithms have emerged as popular alternatives to trust-region and line-search methods for unconstrained optimization. The goal of this research is to tackle some of the challenges associated with cubic regularization and extend the methods to solve constrained problems. The first part of the dissertation is dedicated to enhancing the efficiency of cubic regularization methods for unconstrained optimization without sacrificing their favorable convergence properties. A nonmonotone adaptive cubic regularization approach is proposed that combines adaptive cubic regularization with a line search. In particular, a sufficient decrease in the objective function is obtained by performing a nonmonotone line-search based on satisfying certain strong Wolfe conditions. This is an alternative to repeatedly solving the cubic subproblem with varying regularization parameters and requires lower computational cost and fewer iterations. In addition, two hybrid algorithms are developed that substitute cubic regularization with Newton's method when the objective function is well-behaved or a sufficient descent direction is readily obtainable from a conventional Newton method. By judiciously determining when to utilize cubic regularization, the efficiency of Newton's method can be balanced against the robustness of cubic regularization. In the second part of the dissertation, a novel primal-dual interior-point method for general nonlinear constrained optimization is proposed that minimizes a sequence of shifted primal-dual penalty-barrier functions. The method combines cubic regularization and a line-search to ensure global convergence. The proposed method calculates the cubic regularized step by factoring a sequence of matrices with diagonally-modified primal-dual structure, enabling the use of off-the-shelf linear equation software. This approach allows the extension of the method to large-scale problems while maintaining computational efficiency. Finally, the performance of the proposed algorithms for both unconstrained and constrained optimization is illustrated by extensive numerical results obtained from problems in the CUTEst test collection.