The convergence of matrices generated by rank-2 methods from the restricted β-class of Broyden

SummaryIt is shown that the matricesBk generated by any method from the restricted β-class of Broyden converge, if the method is applied to the unconstrained minimization of a functionf∈C2(Rn) with Lipschitz continuous∇2f(x) and if the method is such that it generates vectorsxk converging sufficiently fast $$(\sum\limits_k {||x_k - x^* ||< \infty } )$$ to a local minimumx* off with positive definite∇2f(x*). This result not only holds for constant step sizesσk≡1 in each iterationxk→xk+1=xk−σkBk−1∇f(xk) of these methods but also for step sizes determined by asymptotically exact line searches. The paper generalizes corresponding results of Ge Ren-Pu and Powell [6] for the DFP and BFGS methods used in conjunction with step sizesσk≡1.