Large-scale eigenvalue problems in trust-region calculations

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 illposed problems. In recent years, methods and software for large-scale trust-region subproblems have been developed that require the solution of a large bordered eigenvalue problem at each iteration. In this work, we describe the bordered eigenvalue problems, the computational challenges in solving them, and present some approaches for their efficient solution by means of Krylov subspace methods for linear and nonlinear eigenvalue problems.