Reverse engineering the objective function

Question

If there is a finite iteration algorithm can we find a function that this algorithm optimizes, in hindsight?

Edit:

Suppose there is a set of functions $f_i(x)$, where $x\in \mathbb R^n$, $i=\{1,\dots,N\}$ and the algorithm works on these function. Can we find a $g(f_1,\dots,f_N)$ up to a constant, such that the algorithm was the optimal solution method for $g()$?

Answer 1

No, because for every objective function $f(x)$ you can find infinitely many other functions that have the exact same maxima, e.g., $g(x)=f(x)+1$.