Superlinear convergence of Broyden's boundedθ-class of methods

Abstract

In this paper the so-called Broyden's boundedθ-class of methods is considered. It contains as a subclass Broyden's restrictedθ-class of methods, in which the updating matrices retain symmetry and positive definiteness. These iteration methods are used for solving unconstrained minimization problems of the following form:$f(\hat x) = \min _{x \in R^n } f(x)$ It is assumed that the step-size coefficientτ _k = 1 in each iteration and the functionalf : R ⁿ → R¹ satisfies the standard assumptions, viz.f is twice continuously differentiable and the Hessian matrix is uniformly positive definite and bounded (there exist constantsm, M > 0 such that m∥y∥² ⩽ 〈y,$m\parallel y\parallel ^2 \leqslant \left\langle {y,f(\hat x)\left. y \right\rangle } \right. \leqslant M\parallel y\parallel ^2$ for ally ∈ R ⁿ) and satisfies a Lipschitz-like condition at the optimal point$\hat x$, the gradient vanishes at$\hat x$ Under these assumptions the local convergence of Broyden's methods is proved. Furthermore, the Q-superlinear convergence is shown.