I am trying to understand descent methods I understand that we move towards the minimum taking steps let's call it $t^{(k)}$ at the current iteration. So the update equation becomes this:
$x^{(k+1)} = x^{(k)}+ t^{(k)}*\Delta{x^{(k)}}$
however, what is the reason we define step t in this way, why do we map it into a linear function as presented below? How can we demonstrate it in a coordinate system?
$t^{(k)} = argmin_{t\leq{0}}{(f(x^{(k) } +\Delta{x^{(k)}}*t))}$
Thanks!