Optimum (min/max) of a symmetric function

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Discussion Overview

The discussion revolves around the properties of symmetric functions, specifically regarding the conditions under which their optimum (minimum or maximum) values occur. Participants explore whether such optima must occur at points where all variables are equal and consider implications for convexity and uniqueness of these optima.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that the optimum value of a symmetric function should occur at the point where all variables are equal, suggesting that symmetry implies this condition.
  • Another participant clarifies that if the optimum exists, it should be unique, leading to the conclusion that all variables must be equal to achieve this uniqueness.
  • A different participant introduces the concept of convexity, noting that for convex functions, local optima are global optima, but questions whether the function in discussion is convex.
  • One participant provides a counterexample involving a function with two global maxima that are not located at equal variable values, challenging the initial argument about symmetry and optima.
  • Further, a participant inquires about the implications of restricting the function to be symmetric and even, questioning the possibility of commenting on its global optimum and convexity.
  • Another participant describes their specific function as a square of a symmetric, bounded, and differentiable function, seeking insights into its convexity and properties related to its global optimum.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of equal variable values for optima in symmetric functions. While some support the idea, others provide counterexamples that challenge this notion, indicating that the discussion remains unresolved.

Contextual Notes

Participants acknowledge the potential limitations of their arguments, including the dependence on the convexity of the function and the specific nature of symmetry. The discussion also highlights the need for further exploration of mathematical properties without reaching definitive conclusions.

Who May Find This Useful

Readers interested in the mathematical properties of symmetric functions, optimization theory, and convex analysis may find this discussion relevant.

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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
 
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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.
 
@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
 
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" 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:
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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