Global convergence properties of two modified BFGS-type methods (Q874351)

The unconstrained optimization problem to find \(\min f(x)\), \(x\in \mathbb R^n\), where the objective function \(f\) is continuously differentiable and satisfies a Lipschitz condition is considered. For a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm based on the new quasi-Newton equation \(B_{k+1}s_k= y_k\), it is proved that the average performance of two of those algorithms is better than that of the classical one. In the presented paper this result is completed with the proof of global convergence of these algorithms associated to a general line search rule.