Given some (convex, I guess) optimization problem, I am told that if it is of the form (max$\{f : \text{insert constraints here}\}$), then we can replace it with an 'equivalent' minimization problem, which is (min$\{-f : \text{same constraints as before}\}$). I understand that if $x$ optimizes the max problem, then it also optimizes the min problem, and vice-versa. However, my question is about the term 'equivalent'. It seems like in general, the objective functions of the max and min problems will not be equal, and hence the feasible solution outputs will not be equal.
My initial assumption was that 'equivalent' problems meant that they were in fact identical, up to formatting (meaning the max f vs. min -f), but this is clearly not the case. I still see how the problems are completely determined by each other, but this was still not what I had interpreted equivalent problems to be.
So, what is the definition of 'equivalent' convex optimization problems, and is there another term that we use when two problems are completely identical up to formatting, or is this not actually something we need to bother considering/defining?