How can I find the minimum value for the sum of absolute values?

[Jorge]

Hello,


I have been having problems finding the way to minimize the sum of absolute values. Specificaly I am looking for the value of X that will minimize the sum|Xi-V|<-- i=1,...n . I know that V should be equal to the mean value of X. But I do not know the correct approach to finding this minimum.

Can I square the Xi-V and differentiate? or is there another approach?

Thanks...

 

[0rthodontist]

So is Xi X times i?

 

[Jorge]

Sorry about that,

i---> is the sub index. Meaning X1...Xn.

Then it is Sum from i={1 to n }of |Xi-V|.

 

[0rthodontist]

Well then that expression is completely independent of the value of X. Do you mean you are looking for a V to minimize that expression? If you are then you can let V be any value between X(n/2) and X(n/2+1) if n is even, and you can let V be X((n+1)/2) if n is odd.