Lately, I have been dealing with a problem that I didn't know how to name it to solve it properly.
The problem is as follow: let's assume that we have a set of elements $A$. And, we have two functions $f$ and $g$, where for any sub-set $B \subset A$ where $\left|B\right| < k$, $k$ is a constraint:
- $f(B)$ : estimates the gain obtained by the set $B$.
- $g(B)$ : estimates the loss obtained by the set $B$.
In our problem we have two strategies $S_1$, $S_2$ which depend on the circumstances of the environment
- $S_1$: selects a set $B_1$ that maximizes the gain
- $S_2$: selects a set $B_2$ that minimizes the loss
my strategy is a hybrid strategy selecting sets $B_1$ and $B_2$ where $\left|B_1\right|+\left|B_2\right|<k$, and we aim to maximize the gain and minimize the loss at the same time. My problem would be: $(max(f(B_1)), min(g(B_2))) s.t. \left|B_1\right|+\left|B_2\right|<k$
NT: given that there are several circumstances sometimes $S_1$ works more efficiently, and some cases $S_2$ works better
Is there anyone who knows what type of problem? Any documentation about it? Since it is an NP-hard problem, is there a way to find an approximation with in the optimal solution?