ABSTRACT

In this paper, we analyse the worst-case complexity of nonlinear stepsize control (NSC) algorithms for solving convex smooth unconstrained optimization problems. We show that, to drive the norm of the gradient below some given positive ε, such methods take at most $\mathcal{O}\left({ϵ}^{-1}\right)$ iterations, which shows that the complexity bound for these methods is in parity with that of gradient descent methods for the same class of problems. As NSC algorithm is a generalization of several methods such as trust-region and adaptive cubic with regularization methods, such bound holds automatically for these methods as well.

中文翻译：

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关于凸光滑无约束优化的非线性步长控制方法的最坏情况复杂性的注释

摘要

在本文中，我们分析了求解凸光滑无约束优化问题的非线性步长控制 (NSC) 算法的最坏情况复杂度。我们表明，为了将梯度的范数驱动到某个给定的正 ε以下，这些方法最多需要 $\mathcal{○}\left({\epsilon }^{-1}\right)$迭代，这表明这些方法的复杂性界限与同一类问题的梯度下降方法相当。由于 NSC 算法是几种方法的推广，例如信任区域和具有正则化方法的自适应三次，因此这种界限也自动适用于这些方法。