Computational and Sensitivity Aspects of Eigenvalue-Based Methods for the Large-Scale Trust-Region Subproblem - extended version

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.Electrical Engineering, Mathematics and Computer Scienc

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.Electrical Engineering, Mathematics and Computer Scienc

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.Electrical Engineering, Mathematics and Computer Scienc