Let's say I have a function in three variables, $f(x, y, z)$. If I wanted to denote the $x$ that maximizes some $f$ for a given $y$ and $z$, I could denote that as

$$ X(y, z) = \mathop{\underset{x}{\arg\max}} f(x, y, z).$$

But what if I maximize jointly over $x$ and $y$ for a given $z$? How would I denote that correctly -- would the following make sense? $$ X(z), Y(z) = \mathop{\underset{x,y}{\arg\max}} f(x, y, z)$$ It appears less technically correct than I want to (or is it obvious that the ordering of the $X$ $Y$ assigns them to the different variables to maximize over in $\max_{x, y}$?

  • $\begingroup$ I think this looks okay. $\endgroup$ Aug 20, 2020 at 17:16
  • 2
    $\begingroup$ I would write $$(X(z),Y(z))=\mathop{\underset{(x,y)}{\arg\max}}f(x,y,z).$$ In other words, one may regard the pair $(x,y)$ as a 'vector-valued single variable' and then apply the idea of $\arg\max$ mutatis mutandis. $\endgroup$ Aug 20, 2020 at 17:29


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