On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Moré and Sorensen, methods which 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 in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems. 1 This work was supported by the EPSRC grants EP/E053351/1 and EP/F005369/1. 2 Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU. Email: [email protected], [email protected] . Current reports available from “http://www.numerical.rl.ac.uk/reports/”. 3 Oxford University Computing Laboratory, Numerical Analysis Group, Wolfson Building, Parks Road, Oxford, OX1 3QD, England, EU. Email: [email protected], [email protected] . Current reports available from “http://web.comlab.ox.ac.uk/oucl/publications/natr/”. Computational Science and Engineering Department Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX February 11, 2009 On solving trust-region and other regularised subproblems in optimization 1