# How to maximize $f$ while minimizing $g$ at the same time?

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:

1. $$f(B)$$ : estimates the gain obtained by the set $$B$$.
2. $$g(B)$$ : 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

1. $$S_1$$: selects a set $$B_1$$ that maximizes the gain
2. $$S_2$$: selects a set $$B_2$$ that minimizes the loss

my strategy is a hybrid strategy selecting sets $$B_1$$ and $$B_2$$ where $$\left|B_1\right|+\left|B_2\right|, and we aim to maximize the gain and minimize the loss at the same time. My problem would be: $$(max(f(B_1)), min(g(B_2))) s.t. \left|B_1\right|+\left|B_2\right|

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?

• You are looking for the concept of "pareto optimal". – Sasha the Noob Jan 13 at 5:40
• What do you mean by "optimal solution" ? Basically, you cannot maximize/minimize more than one variable. You have to define a fitness function, thas is to say a compromise, how much regret do you accept on $f$ to reduce how much $g$ ? Maybe you are just trying to maximize $f-g$ for instance... – Optidad Jan 13 at 9:53