= Solution
The <backtracking line search> starts with $\alpha_{\mathrm{init}}>0$, tests the <Armijo condition>, and repeatedly replaces a failed step by $\tau\alpha$, with fixed $\tau\in(0,1)$. It accepts the first step that passes. Backtracking makes an overambitious trial step small enough for a local linear model to predict the change reliably; part (ii) ensures finite termination. The <Armijo condition> guarantees a decrease of at least $-\alpha\beta g^Tp>0$ along a nonzero <descent direction>. Merely obtaining some decrease without this quantitative lower bound would not provide the convergence estimate in the following parts.
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