A new approach to symmetric rank-one updating

A stabilized version of the symmetric rank-one updating method for solving unconstrained optimization problems is developed by introducing a scaling parameter to ensure that successive estimates of the inverse Hessian are positive definite. The properties of this update are studied, and a new algorithm based on this procedure is proposed. This algorithm uses Davidon's idea of optimal conditioning in order to devise heuristics for selecting the scaling parameter automatically. Numerical testing shows that the new method compares favourably with good implementations of the BFGS method. Thus it appears very competitive in the class of methods which use only function and gradient information.