The problem $$\min \ x^3 \ \mathrm{s.t.} \ x \geq 0$$ is sometimes said to be a convex optimization problem. $f(x) = x^3$ is not a convex function. However, in the domain of $x\geq 0$ it is convex. So for some definitions this is a convex optimization problem, for others it is not. Could you please help me figure out in which main (books like convex analysis, convex optimization, etc.) source this problem is said to be convex?
I am looking to Rockafellar's convex analysis book, it says the objective function needs to be a proper convex function with domain $\{x \ : x \geq 0\}.$ Well, $x^3 $ is not a proper convex function, and has domain $\{x\ : \ x\geq 0 \}$, however $x^3$ restricted to this domain is a convex function. Which interpretation is correct?
Note: the given example is just a simplified version of a higher-level question, so I don't actually care about minimizing $x^3$.