Computational and sensitivity aspects of eigenvalue-based methods for the large-scale trust-region subproblem

The trust-region subproblem of minimizing a quadratic function subject to a norm constraint arises in the context of trust-region methods in optimization and in the regularization of discrete forms of ill-posed problems, including non-negative regularization by means of interior-point methods. A class of efficient methods and software for solving large-scale trust-region subproblems is based on a parametric-eigenvalue formulation of the subproblem. The solution of a sequence of large symmetric eigenvalue problems is the main computation in these methods. In this work, we study the robustness and performance of eigenvalue-based methods for the large-scale trust-region subproblem. We describe the eigenvalue problems and their features, and discuss the computational challenges they pose as well as some approaches to handle them. We present results from a numerical study of the sensitivity of solutions to the trust-region subproblem to eigenproblem solutions.