Solving the Trust-Region Subproblem using the Lanczos Method

The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widely-used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust-region boundary. In this paper, we investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trust-region problem within the currently generated Krylov subspace has very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL-VF05 within the Harwell Subroutine Library. Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 OQX, England, E U Email : [email protected] Current reports available by anonymous ftp from joyous-gard.cc.rl.ac.uk (internet 130.246.9.91) in the directory “pub/reports”. Dipartimento di Informatica e Sistemistica, Universith di Roma “La Sapienza” via Buonarroti 12 00185, Roma, Italy Email : [email protected] , [email protected] Department of Mathematics, FacultCs Universitaires ND de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium, EU Email : [email protected] Current reports available by anonymous ftp from thales.math.fundp.ac.be (internet 138.48.20.102) in the directory “pub/reports”. Department for Computation and Information