On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems that arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Moré and Sorensen, methods that obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS{\tt TRS} and RQS{\tt RQS} as part of the GALAHAD optimization library, and is especially designed for large-scale problems.