Suppose we have a vector valued smooth function $F(x) = [f_1(x),f_2(x),f_3(x),...,f_n(x)]^t$ defined for $x \in \mathbb{R}^n$. Suppose we are given a rectangular domain $M = [a_1,b_1] \times [a_2,b_2] \times ... \times [a_n,b_n]$. Let K be the smallest closed cone containing $F(M)$. If we are given $F$ and $M$, how do we go about computing K?
For example, if K is generated by finitely many vectors, how do we go about computing a generating set of vectors for K? In particular, I'm interested in an algorithm that would take $F$ and $M$, and produce extremal vectors of the cone $K$.
I tried tackling this for the case n = 2. I tried optimizing f_1/f_2 in this case for f_2 positive or negative, although this feels inelegant. I'm also not sure how it would generalize to higher dimensions.