论文标题
经典准牛顿方法的超线性收敛速率
Rates of superlinear convergence for classical quasi-Newton methods
论文作者
论文摘要
我们研究了经典准牛顿方法的局部收敛，以进行非线性优化。尽管很久以前就已经确定了这些方法会渐近地收敛，但相应的收敛速率仍然未知。在本文中，我们解决了这个问题。我们获得了标准的准Newton方法的首次显式非质量速率，该方法基于凸Broyden类的更新公式。特别是，对于众所周知的dfp和bfgs方法，我们获得了$（\ frac {n l^2} {n l^2} {μ^2 k}）^{k/2} $和$（\ frac {n l} {n l} {μk} {μk}）^k/k/2} $的$ k $ n是$ n是$ nisosor，强凸力参数和$ l $是梯度的Lipschitz常数。
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 $(\frac{n L^2}{μ^2 k})^{k/2}$ and $(\frac{n L}{μk})^{k/2}$ respectively, where $k$ is the iteration counter, $n$ is the dimension of the problem, $μ$ is the strong convexity parameter, and $L$ is the Lipschitz constant of the gradient.
