 Solution

Write $\varphi (\alpha )=f(x+\alpha s)$ and $h(\alpha )=\varphi (\alpha )-\varphi (0)-{\beta }_1\alpha d$. Part (a) gives $h(\alpha )<0$ just to the right of zero. If $L$ is a lower bound for $f$, thenbecause $d<0$. By continuity, there is a first positive zero $A$ of $h$, and $h(\alpha )\le 0$ on $[0,A]$. Apply the mean value theorem to $\varphi$ on this interval: for some $t\in (0,A)$,At this same $t$, $h(t)\le 0$ supplies the Armijo condition. Thus $\alpha =t>0$ satisfies both weak Wolfe conditions. The boundedness assumption prevents the objective function from following its initial negative slope forever; this is the mechanism behind Existence of a Wolfe step.