Optimization of several cost functions together

Question

Say I want to minimize several functions together:

$$\min \lVert f_1\rVert, \min \lVert f_2\rVert, \min \lVert f_1-f_2\rVert$$

where $\lVert f\rVert$ is the $L_2$ norm of $f$. I am wondering can I simply find some equivalent process, like

$$\min(a\lVert f_1\rVert+ b \lVert f_2\rVert+ c \lVert f_1-f_2\rVert)$$ ?

If so, is there any relations within $a$, $b$ and $c$? such as $a + b + c = 1$?

Any suggestion is appreciated. Thanks.

Answer 1

There is no general way to do multiobjective optimization.

See e.g. Wikipedia: "For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective."