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.

我们研究经典拟牛顿法的非线性优化问题的局部收敛性。尽管很早以前就已经确定这些方法渐近地渐近收敛，但是相应的收敛速度仍然未知。在本文中，我们解决了这个问题。我们获得标准准牛顿法的超线性收敛的第一个显式非渐近速率，该速率基于凸Broyden类的更新公式。特别是，对于众所周知的DFP和BFGS方法，我们获得的格式为\（（\ frac {n L ^ 2} {\ mu ^ 2 k}）^ {k / 2} \）和\（ （\ frac {n L} {\ mu k}）^ {k / 2} \），其中k是迭代计数器，n是问题的维数，\（\ mu \）是强凸度参数，L是梯度的Lipschitz常数。