I am dealing with a complicated optimization problem with two parameters $N, K$. Setting $K =$ constant and finding the minimum for different $N$'s lead to a set of function values. I can do this for $K = 2,3,\cdots,8$ and $N = 5,6,\cdots,40$. Larger $K$'s or $N$'s take more time to calculate (e.g. 1 hour).
I've plotted what I get (blue) and a lower bound (green) for it, which is described by $L(N,K) = \sqrt{\frac{N-K}{K(N-1)}}$
$K = 3$
$K = 4$
Any of these two are desired :
An expression with respect to $(N,K)$ which best approximates the values I'm getting.
A tighter lower bound (for larger $N$'s)
What methods do you suggest?