Abstract
We present a new matrix-free method for the large-scale trust-region subproblem, assuming that the approximate Hessian is updated by the L-BFGS formula with m=1 or 2. We determine via simple formulas the eigenvalues of these matrices and, at each iteration, we construct a positive definite matrix whose inverse can be expressed analytically, without using factorization. Consequently, a direction of negative curvature can be computed immediately by applying the inverse power method. The computation of the trial step is obtained by performing a sequence of inner products and vector summations. Furthermore, it immediately follows that the strong convergence properties of trust region methods are preserved. Numerical results are also presented.
Acknowledgements
We would like to thank an anonymous referee for helpful suggestions and useful comments that significantly improved the presentation of this work.