I'm studying an optimization book and there's something I can't figure out. It's about the steepest descent method, but I think it's a question that is related to other methods as well. The book says
Consider the ideal case, where the function is quadratic and the line searches are exact.
The function is $$ f(x) = \frac 12 x^T Q x - b^T x$$ where $Q$ is symmetric and positive definite.
Then it says that to compute the step length $a_{k}$ that minimizes $f(x_{k} - a\nabla f_{k})$, we differentiate this function $f(x_{k} - a\nabla f_{k})$ and we set the derivative to zero.
What I don't understand, if we do that, aren't we trying to find the minimizer? If yes, then why go to the trouble of having a line search method. We want to find the minimum of $f$, just differentiate, set equal to $0$ and find the min/max. If we have a method and in every step we set the $\nabla f$ equal to $0$, what are we doing exactly ? Finding all the minimums or ..?
Thanks in advance !