Notes on Limited Memory Bfgs Updating in a Trust{region Framework

The limited memory BFGS method pioneered by Jorge Nocedal is usually implemented as a line search method where the search direction is computed from a BFGS approximation to the inverse of the Hessian. The advantage of inverse updating is that the search directions are obtained by a matrix{ vector multiplication. Furthermore, experience shows that when the BFGS approximation is appropriately re{scaled (or re{sized) at each iteration, the line search stopping criteria are often satissed for the rst trial step. In this note it is observed that limited memory updates to the Hessian approximations can also be applied in the context of a trust{region algorithm with only a modest increase in the linear algebra costs. This is true even though in the trust{region framework one maintains approximations to the Hessian rather than its inverse. The key to this observation is the compact form of the limited memory updates derived by Byrd, Nocedal, and Schnabel. Numerical results on a few of the MINPACK-2 test problems indicate that an implementation that incorporates re{scaling directly into the trust{region updating procedure exhibits convergence behavior comparable to a standard implementation of the algorithm by Liu and Nocedal.