Backtracking Line Search - Graphical Interpretation

Question

I am new to convex optimization and got a little bit confused while reading up on the backtracking line search.

What is $f(x) + \alpha t \nabla f(x)^T\Delta x$ ? I know of the Taylor first order expansion but this does not look like it.

Answer 1

The first order taylor approximation of the function $f(x+t\Delta x)$ at $t=0$ is as follows.

Clearly, derivative of $f(x+t\Delta x)$ with respect to t is $f'(x+t\Delta x) \times \Delta x$ by the chain rule. We need this at $t=0$

Hence, $ f(x) + \ t \nabla f(x)^{T} \Delta x$ is the linear approximator at the point x to the function $f(x+t\Delta x)$

$ f(x) + \alpha \ t \nabla f(x)^{T} \Delta x$ is this line, but with the slope scaled down by $\alpha \in (0,0.5) $