I need to solve a problem to choose $K$ elements from a set of size $N$. The objective function to minimize is highly nonlinear and rather complicated, so picking one element might change the "value" of all other elements significantly.
My general question is: Is there anything I can do aside from brute forcing through all combinations? I'd normally use heuristic methods for this, but I need a verified optimal solution this time.
I've tried to identify some elements that are not part of the solution beforehand, or find some upper/lower bound on the value of each element, but no luck so far.
If anyone wants to try I'll write the objective function down, but I don't think there's an easy solution to this.
I am given an $n\times n$ matrix $A$ is symmetric and has only positive elements on the main diagonal. In fact $A = (S^T S)^{-1}$.
Choose $k$ integers $\vec{t}=[t_1,...,t_k]\in\mathbb{N}$, $1\leq t_i\leq n$ such that: $$\min\limits_\vec{t}(||diag(A)||_2)\qquad A\in\mathbb{R}^{n\times n}$$
Where every $t_i$ added (iteratively) changes every element of the matrix $A$ by $$a_{x,y,t_{i}}=a_{x,y}-\frac{a_{x,t}a_{t,y}}{1+a_{t,t}}$$