# Suppose we have some functions f 1 </msub> ( x ) , f 2

Suppose we have some functions ${f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{n}\left(x\right)$ with $x\in {\mathbb{Z}}^{n}$.

We can denote the subset ${X}_{1}$ of ${\mathbb{Z}}^{n}$ that maximizes ${f}_{1}\left(x\right)$ as:
${X}_{1}=\underset{x\in {\mathbb{Z}}^{n}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}\left(x\right)$

Now, suppose there is a kind of "priority" in which I also want to maximize ${f}_{2}$, as long as I keep maximizing ${f}_{1}$. This could be represented as:
${X}_{2}=\underset{x\in {X}_{1}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{2}\left(x\right)$

The same for ${f}_{3}$:
${X}_{3}=\underset{x\in {X}_{2}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{3}\left(x\right)$

So on and so forth...

Is there some, more concise, notation to represent this "maximization priority"?
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Timothy Mcclure
I figured it out and decided to post an answer in case more people are interested.

This maximization with priority is indeed a type of multi-objective optimization. More specific it is what is called the lexicographic method. In the lexicographic method, functions are arranged in priority. The formal definition is as follow:
$\begin{array}{rlrl}& \underset{\mathbf{x}\in \mathbf{X}}{\text{minimize}}& & {f}_{l}\left(\mathbf{x}\right)\\ & \text{subject to}& & {f}_{j}\left(\mathbf{x}\right)\le {\mathbf{y}}_{j}^{\ast },\phantom{\rule{thickmathspace}{0ex}}j=1,\dots ,l-1,\end{array}$
where ${\mathbf{y}}_{j}^{\ast }$ is the optimum of the $j$th objective function with $l=j$.

Now going back to the question, we may adapt the lexicographic optimization to fit in the "arg max" problem, it becomes:
$\begin{array}{rlrl}& \underset{x\in {\mathbb{Z}}^{n}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}& & {f}_{l}\left(x\right)\\ & \text{subject to}& & {f}_{j}\left(x\right)\le {\mathbf{y}}_{j}^{\ast },\phantom{\rule{thickmathspace}{0ex}}j=1,\dots ,l-1,\end{array}$