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Confusing max(min(f(x,y)) notation?

  1. Feb 17, 2014 #1
    I'm working on some MIT OCW for probability theory, and I've come across some confusing notation in the assignmnet. Look at exercise 3 here:

    http://ocw.mit.edu/courses/electric...y-fall-2008/assignments/MIT6_436JF08_hw01.pdf

    I understand what the max and min functions do on their own (find the max or min value of the function subject to the constraints written under the function), but what does it mean when they are combined like this?

    Are they applied one after the other? So that max min f(x,y) would mean to find the minimum function f(x) (whatever that means, there is no given measure of the "size" of a function) then find the maximum value of the function f(x). Or are they somehow applied together?

    Perhaps [tex]min_{y \in B} f(x,y)[/tex] means to find the function f(x) (set y constant) such that f(x) has the smallest minimum value (the global min)? likewise, [tex]max_{x\in B} f(x,y)[/tex] means to find the function f(y) (set x constant) such that f(y) has the largest maximum value (the global max)? I don't think this is right though, since it is easy to imagine a counter example for the statement we are supposed to prove.

    Anyone know exactly what this notation means?

    Thanks for any help.
     
    Last edited: Feb 17, 2014
  2. jcsd
  3. Feb 17, 2014 #2

    AlephZero

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    ##\min_{y \in B}f(x,y)## is a function of ##x##. For a given (fixed) value of ##x##, it is the smallest value of ##f(x,y)##, for any value of ##y \in B##.

    ##\max_{x \in A} \min_{y \in B} f(x,y)## means ##\max_{x \in A} \left( \min_{y \in B} f(x,y)\right)##.
     
  4. Feb 17, 2014 #3
    I understand everything in the first line of your answer.

    But, I still don't understand completely. Why are there no quantifiers or anything in the problem statement? I don't see how the definition you have given can lead to a sensible interpretation of the problem. If you have some fixed x for the left side of the ≤ sign, what do you do with it on the right side? It loses it's meaning when you go to the right side of ≤.
     
  5. Feb 17, 2014 #4
    For [itex]A,B[/itex] finite and [itex]f:A\times B\to\mathbb R[/itex], we're interested in properties of the number [itex]\alpha = \max_{x\in A}\min_{y\in B}f(x,y)[/itex].
    -To start, define the function [itex]g:A\to\mathbb R[/itex] via [itex]g(x) = \min_{y\in B} f(x,y)[/itex]. This is a well defined number. Indeed, for any given [itex]x\in A[/itex], the right-hand side expression is a precisely defined number.
    -Now, [itex]\alpha[/itex] is defined similarly, just as [itex]\alpha=\max_{x\in A}g(x)[/itex].

    [itex][/itex]

    Now, [itex]\beta = \min_{y\in B}\max_{x\in A}f(x,y)[/itex] is a (possibly different) number too, and so it makes sense to ask whether or not [itex]\alpha\leq\beta[/itex].
     
  6. Feb 18, 2014 #5
    I see. Thank you :)
     
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