I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such that maximize A will somewhat minimize B and wise-versa. So that both A and B can't be optimize both at the same time.

I have browsed around the web, one good solution to my current situation is to do "weighted optmization", such that introudcing an additional parameter $w$.

The optimization problem now becomes weighted importance of the two. So now, the new objective function is :

$w\*A+(1-w)\*B$

With this, the goal now is the maximize above.

But I found the above equation is extremely difficult to solve.(Running CAD tool for several hours and it can't give anything, and I try other method , and still nothing works). I am wondering if there is better "weighted optimization" format (for example, above is the sum of the two), so that I can try and see if I can get something?

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Update: sorry, I know this is a very general problem. But I am new to this area. If you guys can just give me a keyword to google, or point me to the right direction, that is good enough.

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Update:

Below is the image of the problem:
as you can see A is parabolic,
B is like a staight line. I am plotting the case when the weighting factor $w=0.7$
![enter image description here](https://i.stack.imgur.com/xGNaQ.png)