The superlinear convergence of a modified BFGS-type method for unconstrained optimization

The BFGS method is the most effective of the quasi-Newton methods for solving unconstrained optimization problems. Wei, Li, and Qi [6] have proposed some modified BFGS methods based on the new quasi-Newton equation B(k+1)s(k)=y(k)*, where y(k)* is the sum of y(k) and A(k)s(k), and A(k) is some matrix. The average performance of Algorithm 4.3 in [16] is better than that of the BFGS method, but its superlinear convergence is still open. This article proves the superlinear convergence of Algorithm 4.3 under some suitable conditions.