  It has long been known that the gradient (steepest descent) method may fail
on nonsmooth 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 nonsmooth
convex function $f(x) = a|x^{(1)}| + \sum_{i=2}^{n} x^{(i)}$. 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 $x_0 \in
\R^n$ with $x^{(1)}_0\not = 0$, the iterates converge to a point $\bar x$ with
$\bar x^{(1)}=0$, although $f$ is unbounded below. We also give conditions
under which the iterates $f(x_k)\to-\infty$, using a specific Armijo-Wolfe
bracketing line search. Our experimental results demonstrate that our analysis
is reasonably tight.
