Exact line search in convex optimization

Question

For a convex function $f$ what do we know about convexity of the exact line search problem?

$$\min_{\alpha \ge 0} f(x+ \alpha p_k)$$

I think because the function is convex and is linear in variable, and constraint is linear, the exact line search is convex in variable so it has a global minimum as opposed to local min. Want to make sure that is true.

Answer 1

Your intuitions are right. Indeed, $g: \alpha \mapsto x + \alpha p_k$ is affine whilst $f$ is convex. Therefore $f \circ g$ is convex.