Sum of squares of differences of functions

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Discussion Overview

The discussion revolves around the relationship between the minimum of the sum of squares of differences of functions and the minimum of the product of squares of differences of functions. It explores whether solutions that maximize one functional also maximize the other, under the condition that no difference is equal to zero.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions if the minimum of the sum of squares of differences is the same as the minimum for the product of squares of differences, assuming no difference equals zero.
  • Another participant requests an example in equation form to clarify the initial question.
  • A further elaboration presents two functionals to maximize: the sum of squares of differences and the product of squares of differences, questioning if solutions for one are also solutions for the other.
  • One participant expresses that the question appears too general to provide a definitive answer.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing views on the generality of the question and the need for examples to clarify the concepts involved.

Contextual Notes

The discussion lacks specific examples or definitions that could clarify the assumptions and scope of the functions involved, which may affect the analysis of the proposed relationships.

brydustin
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If you are looking for a minimum of the sum of squares of differences of functions should it be the same as the minimum for the product of squares of differences of functions?
Also assume that no difference is equal to zero.
 
Last edited:
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Could you write some example of what you mean (in equation form)?
 
mathman said:
Could you write some example of what you mean (in equation form)?
For a set of functions f(X)_i , where i is an index and X is a vector in R^n.

Maximize:
summation_ i=1^N summation_ j = (i+1)^N (f(X)_i - f(X)_ j )^2 => VS
Maximize:
product_ i=1^N product_ j = (i+1)^N (f(X)_i - f(X)_ j )^2

Obviously the values are different, my question is: are there solutions (values X, which maximize the functional, i.e. local maximum, gradient = 0, not a saddle point, etc...) for one that are also solutions for the other (should one have more solutions than the other, i.e. is the set of solutions of one function a proper subset of the other). The actual functions f(x)'s are not important to discuss here, this is an analysis question.
 
The question (to me) looks too general to answer.
 

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