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

measgachyx5q9 2022-05-08 Answered
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"?
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Answers (1)

Timothy Mcclure
Answered 2022-05-09 Author has 12 answers
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 "arg max" 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 ,
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