Exclamation Mark Symbol (Optimization)

Question

I hope my question is not off-topic.

I have recently begun to learn optimization in mathematics and it is common to see notations like:

(1) $ f(x,y) \rightarrow min ! $

(2) $ f(x,y) \rightarrow max ! $

or

(3)

I know that in (1) I am supposed to minimize the function, in (2) to maximize the function and in (3) to set the gradient equal to 0, but I would like to know if the exclamation mark has some special meaning or it’s just a notation.

Thank you in advance.

Answer 1

I think in these cases the exclamation point indicates “to be done”. So in the first case it tells you that the function $f$ should be or will be minimized. In the second, I would expect that $L$ depends on some parameters and that the next part of the script is a calculation that finds values for these parameters such that $\nabla L = 0$. Neither the factorial nor uniqueness is relevant here, I think.

Answer 2

It is relatively common to use an exclamation mark as an abbreviation for the phrases "unique", "uniqueness", "is unique", etc. For instance, it is common to abbreviate "exists unique" as $\exists !$

In this case, I would interpret $(1)$ and $(2)$ as finding the maximizer/minimizer and asserting that it is unique. Similarly, I would interpret $(3)$ as finding the critical point and asserting that it is unique.