Sum of squares of differences of functions

1. Dec 17, 2011

brydustin

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: Dec 17, 2011
2. Dec 17, 2011

mathman

Could you write some example of what you mean (in equation form)?

3. Dec 17, 2011

brydustin

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.

4. Dec 18, 2011

mathman

The question (to me) looks too general to answer.