What's $argmax$ in multiobjective optimization? Is it component-wise or something else?

Question

What's $argmax$ in multiobjective optimization? Is it component-wise or something else?

Such as in the weighting method:

$$\sum_{i=1}^n w_i p_i$$

then what is

$$argmax_p\sum_{i=1}^n w_i p_i$$

Is it the $p$ that produces the largest sum? Or that has some kind of component-wise dominance?

Answer 1

It is the value of $p_1,\ldots,p_n$ (possibly under some constraints) that maximizes $\sum_{i=1}^n w_i p_i$.

Answer 2

$max\{f(x)\}$ is the maximum value the function assumes, while $argmax\{f(x)\}$ is the value of the $x$ on which the function $f$ assumes the maximum value. The first is in the codomain and the latter in the domain