M.S. Apostolopoulou, D.G. Sotiropoulos and P. Pintelas, Solving the quadratic trust-region subproblem in a low-memory BFGS framework, Optimization Methods and Software, Volume 23,  Number 5,  p.p. 651--674, 2008.

 

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.