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I am trying to solve a dual problem. And it is said that min max f() is always smaller or equal to max min f(). For example, $\max_{y \in Y} \min_{x \in X} f(x,y)$ is always smaller or equal to $ \min_{x \in X} \max_{y \in Y} f(x,y) $

My question is ,how is $ \max_{y \in Y} f(x,y) $ properly defined?? Can variable y be eliminated? I hardly agree so... If not, how is it properly defined??

Thank you very much.

Best Richie

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Probably $\max_{y ∈ Y} f(x, y)$ is just a function on $X$ which assigns $x \mapsto \max_{y ∈ Y} f(x, y)$.

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  • $\begingroup$ Thanks for your reply.. Could I ask if the result will contain y or not? I agree that it will be a function of x. But will y be eliminated? $\endgroup$ Nov 21, 2013 at 16:41
  • $\begingroup$ @richieqianle: What do you mean by result containing $y$? The resulting object is just a function of one variable which to some $x$ assigns the value $\max_{y ∈ Y} f(x, y)$ or to some $a$ assigns the value $\max_{b ∈ Y} f(a, b)$. Actually the variables $x$, $y$ exist no more. They are just helper syntactical objects by which we define the function we want. $\endgroup$
    – user87690
    Nov 22, 2013 at 10:43

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