We present a new matrix-free method for the trust region subproblem, assuming that the approximate Hessian is updated by the limited memory BFGS formula with m = 2. The resulting updating scheme, called 2-BFGS, give us the ability to determine via simple formulas the eigenvalues of the resulting approximation. Thus, at each iteration, we can 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. Thus, the computation of the trial step can be 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.
Technical Report No. TR07-01, University of Patras, Department of Mathematics, June 2007. Appeared in: Optimization Methods and Software http://dx.doi.org/10.1080/10556780802413579