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Optimum (min/max) of a symmetric function

  1. Mar 13, 2009 #1
    Hi all,
    I'm wondering if the following argument is right:

    "The optimum (minimum/maximum) value of a symmetric function f(x_1,x_2,...,x_n) (By 'symmetric' I mean that f remains same if we alter any x_i's with x_j's), if exists, should be at the point x_1=x_2=...=x_n".

    Please help me by proving (or disproving, i.e, a counter example) this argument. I have examine some well-known symmetric functions and these obey the rules. However, a general proof (or reference to literature, document etc.) is needed.


    [I think this may be right, in the sense that the graph of the function (in n+1 dimensional space) should be symmetric about all axes, so if f has optimum at x_1=a, by the symmetry of the curve, x_2=...=x_n=a.]


    Regards,
    NaturePaper
     
  2. jcsd
  3. Mar 13, 2009 #2
    I assume that you mean that the optimum, if it exists, should be unique. Say it is some point [tex](a_1,\ldots,a_n)[/tex], but then [tex]a_1=\ldots=a_n[/tex], because otherwise one could interchange two coordinates to obtain another optimum.
     
  4. Mar 14, 2009 #3
    @yyat,
    Oh..., Let us consider the global optimum (min/max) for f. For convex f, it is known that every local optimum is the global optimum. However, I don't know whether f is convex or not. But I think the following discussion is right:

    Let f has a global optimum (by definition, it is unique). Then the set of points where f attains it should contain the point x_1=x_2=...=x_n= some constant, by your argument.

    Am I right?

    Regards
    NaturePaper
     
  5. Mar 14, 2009 #4
    By uniqueness I meant that the optimum is only achieved at a single point, otherwise the argument becomes false. For example, consider the function on R^2 that is mostly 0 but has two http://en.wikipedia.org/wiki/Bump_function" [Broken] at (1,0) and (0,1) that are mirror images of themselves with respect to the diagonal (x_1=x_2). Then the function has two global maxima, but none of them on the diagonal.
     
    Last edited by a moderator: May 4, 2017
  6. Mar 17, 2009 #5
    Anyway, if I restrict f to be symmetric, even function [i.e. f(-X)=f(X)], any number of time differentiable, then is it possible to make any comment about its global optimum?
    At least, is f convex etc...?

    Actually, my function f is like f=[g(x_1,x-2,...,x_n)]^2, where g is symmetric, bounded and any time differentiable. Is it possible to comment about the convexity of f etc?

    Regards,
    NaturePaper
     
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