# In-exact line search

In my class notes, the author says:

"If $$f:\mathbb{R}^n \to \mathbb{R}$$ is bounded below and $$p_k$$ is a descent direction and the $$\alpha-\beta$$ also known as Armijo-Goldstein condition is met then either $$\nabla f(x_k) \to 0$$ or the angle between $$\nabla f(x_k)$$ and $$p_k$$ approaches 90 degrees."

I think the author meant $$\| \nabla f(x_k) \| \to 0$$ as this would imply that we are approaching the stationary point of $$f$$??

or in the worst case scenario $$\nabla f(x_k)$$ and $$p_k$$ becomes orthogonal. It means that $$p_k$$ still decreases the $$f$$ but the rate of decrease decreases?

• If $\nabla f(x_k)\rightarrow 0$, then clearly also $\|\nabla f(x_k)\|\rightarrow 0$. The opposite is in fact also true: The two conditions are equivalent. – Wolfgang Bangerth Jan 21 at 15:22