multi objective optimization

Question

Suppose we want to maximize two positive bounded objectives. A usual approach for this aim is to maximize a weighted sum of these two objectives.

Now, my question is why not to maximize the product of these two objectives? Are there any disadvantages?

Answer 1

Like more complex derivatives, loosing convexity (in case of minimization), loosing concavity (in maximization) ... ?

Also it is the same as optimizing the sum of the logarithm of the objectives.