  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 $(\frac{n L^2}{\mu^2 k})^{k/2}$ and
$(\frac{n L}{\mu k})^{k/2}$ respectively, where $k$ is the iteration counter,
$n$ is the dimension of the problem, $\mu$ is the strong convexity parameter,
and $L$ is the Lipschitz constant of the gradient.
