We have a simple problem, namely minimizing: $$f(x) = x_1^2 + x_2^2 - x_1.$$
The gradient is $$\nabla f(x) = \begin{bmatrix} 2x_1 - 1 \\ 2x_2 \end{bmatrix},$$ hence the unique stationary point is: $x_* = (\frac{1}{2}, 0)$. Now, I have a simple introduction to OR level question asking me why this is globally optimal.
My typical answer is to show that $f(x)$ is convex, e.g.,: $$ \nabla^2 f(x) = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \succ 0,$$ therefore a local minimizer is a global minimizer.
However, in some engineering course-notes, the proof of global optimality is:
It is also a global minimizer since the function $f$ is $C^1$ and radially unbounded, therefore the global minimizer is a stationary point.
What is $C^1$?
What does radial unboundedness imply here?
Can anyone, please, enlighten me?
Edit: I think they are using the fact that $\lim\limits_{k \to \infty} \|x_k\|=\infty \implies \lim\limits_{k \to \infty} f(x_k) = \infty$. Hence, all level sets of $f$ are compact. Hence the global minimum exists in $\mathbb{R}^2$. And the unique stationary point is also globally optimum. But the main question is,
- Why not show convexity and go in this way?