Armijo condition ($f(x+\alpha s)\le f(x)+{\beta }_1\alpha \nabla f(x{)}^{T}s$)

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 $d=\nabla f(x{)}^{T}s<0$. 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.