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 $B \in A$, where $|B|< k$, $k$ 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 depends on the circumstances of the environment
- $S_1$: selects a set $B$ that maximizes the gain
- $S_2$: selects a set $B$ that minimizes the lost
My strategy is a hybrid strategy, which is selecting a sets $B_1$ and $B_2$ where $|B_1|+|B_2|<k$, and we aim to maximize the gain and minimize the lost at the same time.
My problem would be: $(\max\left(f(B_2)\right), \min\left(g(B_2)\right)) \,\mathrm{s.t.}\, |B_1|+|B_2|<k$
NT: given that there are several circumstances some times $S_1$ 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 a NP-hard problem, is there way to find an approximation within the optimal solution?
in a very general way, there wont be an optimal solution for this type of problem. Instead, you will obtain a set of optimal solutions known as frontiers that maximizes $f$ for a given lowest $g$. Try checking out mean-variance optimization in modern portfolio theory. The problem is similar as yours, i.e. maximizes return and minimize volatility