= Armijo condition
{c}
{title2=$f(x+\alpha s)\leq f(x)+\beta_1\alpha\nabla f(x)^Ts$}
= Sufficient decrease condition
{synonym}
For $0<\beta_1<1$ and a <descent direction> $s$, the <Armijo condition> asks for a decrease at least a fixed fraction $\beta_1$ of the decrease predicted by the first-order <Taylor expansion>. Put $d=\nabla f(x)^Ts<0$. <Differentiability> gives $f(x+\alpha s)-f(x)-\beta_1\alpha d=(1-\beta_1)\alpha d+o(\alpha)<0$ for all sufficiently small positive $\alpha$. Thus the condition is satisfiable without a second derivative.
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