Suppose we have some functions f 1 ( x ) , f 2 ( x ) , … , f n ( x ) with x ∈ Z n .We can denote the subset X 1 of Z n that maximizes f 1 ( x ) as: X 1 = a r g m a x x ∈ Z n 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 = a r g m a x x ∈ X 1 f 2 ( x )The same for f 3 : X 3 = a r g m a x x ∈ X 2 f 3 ( x )So on and so forth...Is there some, more concise, notation to represent this "maximization priority"?

Question

Suppose we have some functions       f    1    (  x  )  ,      f    2    (  x  )  ,  …  ,      f    n    (  x  ) with   x  ∈            Z        n  .We can denote the subset       X    1   of             Z        n   that maximizes       f    1    (  x  ) as:      X    1    =            a      r      g            m      a      x              x      ∈                        Z                n                    f    1    (  x  )Now, suppose there is a kind of &quot;priority&quot; in which I also want to maximize       f    2  , as long as I keep maximizing       f    1  . This could be represented as:      X    2    =            a      r      g            m      a      x              x      ∈              X        1                    f    2    (  x  )The same for       f    3  :      X    3    =            a      r      g            m      a      x              x      ∈              X        2                    f    3    (  x  )So on and so forth...Is there some, more concise, notation to represent this &quot;maximization priority&quot;?

Timothy Mcclure · Accepted Answer

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:                                    minimize                                    x                        ∈                          X                                                                  f          l                (                  x                )                                  subject to                                    f          j                (                  x                )        ≤                              y                    j          ∗                ,                j        =        1        ,        …        ,        l        −        1        ,            where             y        j    ∗   is the optimum of the   jth objective function with   l  =  j.Now going back to the question, we may adapt the lexicographic optimization to fit in the &quot;arg max&quot; problem, it becomes:                                                a            r            g                        m            a            x                                x            ∈                                          Z                            n                                                                  f          l                (        x        )                                  subject to                                    f          j                (        x        )        ≤                              y                    j          ∗                ,                j        =        1        ,        …        ,        l        −        1        ,

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

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