On the Behavior of Damped Quasi - Newton Methods for Unconstrained Optimization

We consider a family of damped quasi-Newton methods for solving unconstrained optimization problems. This family resembles that of Broyden with line searches, except that the change in gradients is replaced by a certain hybrid vector before updating the current Hessian approximation. This damped technique modifies the Hessian approximations so that they are maintained sufficiently positive definite. Hence, the objective function is reduced sufficiently on each iteration. The recent result that the damped technique maintains the global and superlinear convergence properties of a restricted class of quasi-Newton methods for convex functions is tested on a set of standard unconstrained optimization problems. The behavior of the methods is studied on the basis of the numerical results required to solve these test problems. It is shown that the damped technique improves the performance of quasi-Newton methods substantially in some robust cases (as the BFGS method) and significantly in certain inefficient cases (as the DFP method).