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.

## Citation

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