Solving the trust-region subproblem using the Lanczos method

Nick Gould, Stefano Lucidi, Massimo Roma, Philippe Toint

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    Abstract

    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 a 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.
    Original languageEnglish
    Pages (from-to)504-525
    Number of pages22
    JournalSIAM Journal on Optimization
    Volume9
    Issue number2
    Publication statusPublished - 1 Mar 1999

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