Rates of superlinear convergence for classical quasi-Newton methods

Abstract

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form $(\frac{n{L}^2}{\hat{I}{¼}^{2}k}{)}^{k/2}$ and $(\frac{nL}{\hat{I}¼k}{)}^{k/2}$ respectively, where $k$ is the iteration counter, $n$ is the dimension of the problem, $\hat{I}¼$ is the strong convexity parameter, and $L$ is the Lipschitz constant of the gradient.