A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem

Marielba Rojas, Sandra A. Santos, and Danny C. Sorensen

Reports in Informatics No. 175, September 1999, Department of Informatics, University of Bergen, Norway.


We present a matrix-free algorithm for the large-scale trust-region subproblem. Our algorithm relies on matrix-vector products only and does not require matrix factorizations. We recast the trust-region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter. We then find the optimal solution of the trust-region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter. The new algorithm uses a different interpolation scheme than existing methods and introduces a unified iteration that naturally includes the so-called hard case. We show that the new iteration is well defined and convergent at a superlinear rate. We present computational results to illustrate convergence properties and robustness of the method.