Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a convex, differentiable function. Suppose that there is a nonempty set of points that minimize $f$. If $\{x_k\}$ is the sequence produced by gradient descent with the Armijo rule, is it always true that $f(x_k) \rightarrow f(x^*)$?
From Proposition 1.2.1 from "Nonlinear Programming" by Bertsekas, we know that if $\{x_k\}$ is a sequence generated with the Armijo rule, every limit point will be a minimizer, so we have that $f(x_k) \rightarrow f(x^*)$. However, in the case where the level set $\{x: f(x) \leq f(x_0)\}$ is not bounded, then it's possible that no limit points exist.