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Suppose we have some functions $f_1(x), f_2(x), \ldots, f_n(x)$ with $x \in \mathbb{Z}^n$.

We can denote the subset $X_1$ of $\mathbb{Z}^n$ that maximizes $f_1(x)$ as:

$$X_1 = \underset{x \in \mathbb{Z}^n}{\operatorname{arg\,max}}\, f_1(x)$$

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:

$$X_2 = \underset{x \in X_1}{\operatorname{arg\,max}}\, f_2(x)$$

The same for $f_3$:

$$X_3 = \underset{x \in X_2}{\operatorname{arg\,max}}\, f_3(x)$$

So on and so forth...

Is there some, more concise, notation to represent this "maximization priority"?

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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:

$$ \begin{aligned} & \underset{\mathbf{x} \in \mathbf{X}}{\text{minimize}} & & f_l(\mathbf{x}) \\ & \text{subject to} & & f_j(\mathbf{x})\leq\mathbf{y}^*_j,\;j=1,\dotsc,l-1,\\ \end{aligned} $$

where $\mathbf{y}^*_j$ 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:

$$ \begin{aligned} & \underset{x \in \mathbb{Z}^n}{\operatorname{arg\,max}} & & f_l(x) \\ & \text{subject to} & & f_j(x)\leq\mathbf{y}^*_j,\;j=1,\dotsc,l-1,\\ \end{aligned} $$

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