Wolfe conditions

For a continuously differentiable function, a descent direction $s$ and $0<{\beta }_1<{\beta }_2<1$, the Wolfe conditions require both the Armijo condition andThis second inequality is the weak curvature condition: the slope must have become less negative. The strong variant instead requires $|\nabla f(x+\alpha s{)}^{T}s|\le {\beta }_2|\nabla f(x{)}^{T}s|$. The weak version already guarantees a positive curvature pair for the Quasi-Newton secant equation.

Suppose $f$ is continuously differentiable, $s$ is a descent direction, and $f(x+\alpha s)$ is bounded below for $\alpha \ge 0$. A positive step size satisfying the weak Wolfe conditions exists. To prove this, put $\varphi (\alpha )=f(x+\alpha s)$, $d={\varphi }^{\prime }(0)<0$, and $h(\alpha )=\varphi (\alpha )-\varphi (0)-{\beta }_1\alpha d$. The Armijo condition proof shows $h<0$ just to the right of zero. Boundedness below gives $h(\alpha )\to +\infty$. Consequently there is a first positive zero $A$, with $h\le 0$ on $[0,A]$. By the mean value theorem, some $t\in (0,A)$ satisfiesAt $t$, $h(t)\le 0$ gives the Armijo condition, and the displayed derivative gives the weak Wolfe conditions.