Zoutendijk theorem (${\sum }_k{\cos }^2{\theta }_k\Vert \nabla f(x_k){\Vert }^2<\infty$)

Suppose $f$ is bounded below on its initial level set, continuously differentiable with a Lipschitz gradient nearby, and each descent direction $p_k$ uses a step satisfying fixed Wolfe conditions. If $\cos {\theta }_k=-\nabla f(x_k{)}^T{p}_k/(\Vert \nabla f(x_k)\Vert \Vert p_k\Vert )$, the displayed series converges. In particular a uniform positive lower bound on these cosines gives $\nabla f(x_k)\to 0$.

For a gradient Lipschitz constant $L$, the curvature condition gives ${\alpha }_k\ge (1-\eta )(-\nabla f^T{p}_k)/(L\Vert p_k{\Vert }^2)$. Sufficient decrease then gives $f(x_k)-f(x_{k+1})\ge \sigma (1-\eta ){\cos }^2{\theta }_k\Vert \nabla f(x_k){\Vert }^2/L$. Summing and using the lower bound on $f$ proves the assertion.