It has long been known that the gradient (steepest descent) method may fail on non-smooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an analysis of the gradient method with steplengths satisfying the Armijo and Wolfe inexact line search conditions on the non-smooth convex function $f\left(x\right)=a|x^{\left(1\right)}|+\sum _{i=2}^n{x}^{\left(i\right)}$. We show that if a is sufficiently large, satisfying a condition that depends only on the Armijo parameter, then, when the method is initiated at any point $x_0\in {\mathbb{R}}^n$ with $x_0^{\left(1\right)}\ne 0$, the iterates converge to a point $\overline{x}$ with ${\overline{x}}^{\left(1\right)}=0$, although f is unbounded below. We also give conditions under which the iterates $f\left(x_k\right)\to -\mathrm{\infty }$, using a specific Armijo–Wolfe bracketing line search. Our experimental results demonstrate that our analysis is reasonably tight.

早就知道梯度（最速下降）方法可能会在非平滑问题上失败，但是文献中出现的示例要么专门设计用于通过精确的线搜索来克服梯度方法或次梯度方法，要么不稳定。关于初始点的扰动。我们对步长满足非光滑凸函数上Armijo和Wolfe不精确线搜索条件的梯度方法进行了分析$F（X）=\mathrm{一种}|X^{（\mathrm{1个}）}|+\sum _{\mathrm{一世}=2}^{ñ}{X}^{（\mathrm{一世}）}$。我们证明，如果a足够大，满足仅取决于Armijo参数的条件，那么在任何时候启动该方法时$X_0\in {\mathbb{[R}}^{ñ}$ 与 $X_0^{（\mathrm{1个}）}\ne 0$，迭代收敛到一点 $\overline{X}$ 与 ${\overline{X}}^{（\mathrm{1个}）}=0$，尽管f在下面是无限的。我们还给出了迭代的条件$F（X_{ķ}）\to -\mathrm{\infty }$，使用特定的Armijo–Wolfe包围线搜索。我们的实验结果表明我们的分析相当严格。