mathematical optimization - Optimizing up to tolerance

Question

I have multi-object optimization problem, the first objective is to minimize F(x) as possible (zero is the best case). The second objective is to minimize G(x) up to a certain level (10 is enough). However, that does not mean I need G(x) to be exactly 10. I need it to be minimized to 10 or less but I am happy with 10.

Currently I am simply minimizing (F(x) + G(x))^2. However, the final result looks like:

F(min_x) = 5 & G(min_x) = 5

I prefer result such as:

F(min_x) = 0.1 & G(min_x) = 9.5

I could achieve sth similar by playing the weight, so I am optimizing:

(10*F(x) + 1*G(x))^2

However, it was not robust and it makes the optimization process slower.. Furthermore choosing the weights is not easy (I could not find weights that works alwayse).

Question:

Is there a defacto way of formalizing the optimization when problem in the case I have explained?

Answer 1

等待大神答复