Is this writing correct

Question

Can I define an optimization problem like this to save some space? $$\min_x f(x)=x^2+1-log(x)\\s.t. ~x>0,$$ or i have to write it as $$\min_x f(x)\\s.t. ~x>0,$$ where $f(x)=x^2+1-log(x)$?

Update: In fact, in the real problem, I have a set of constraints like $\{g_i(x)\ge 0\}_{i=1}^k$, and I need to define them in separate lines or with "," in between, but not as the underset of "$\min$".

Answer 1

My two pence:

$$\min_{x>0}f(x), f(x):=x^2+1-\log x$$

or just

$$\min_{x>0}(x^2+1-\log x)$$

in case you don't need the function elsewhere (or let it be $f(x)$ implicitly).

Answer 2

No, because this writing is ambiguous, because it is unclear whether you are giving the limit of f (not the second expression) or if you ask the limit of the function (like in the second expression).

I recommend the following:

$\min_{x>0} f(x)=\min_{x>0}(x^2+1-log(x))$

Answer 3

For $f(x)=x^2+1-\ln{x}$ I like the following. $$\min_{(0,+\infty)}f.$$