# absolute value (positive and negative part)

I found a notation that $$|y_i| = y_i^{+}+y_i^{-}$$ where y is y n dimensional vector.

what does +/- imply? I understand that we can have both negative and positive value into absolute function, but eventually we will always have the positive outcomes. how can absolute value be equivalent to sum of negative value and positive value?

That is generally used with the meaning $y_i^+=\max(y_i, 0)$ and $y_i^-=\max(-y_i, 0)$, and the equality holds. Does it make sense for you?
• It is not like you have a positive and negative part. If $y_i$ is positive, note that $y_i^-$ is $0$. If on the contrary $y_i$ is negative, then $y_i^+$ is $0$. Now any number $a$ admits the decomposition $a=a+0$ – user194469 Apr 24 '16 at 23:29
• Ok. I am trying to understand following equivalence, [y = m - v, m > 0, v > 0] <=> $$(y_i^+)+(y_i^-) = |y_i| <= p$$. Even if I understand the meaning of decomposing the absolute value, the equivalent statement is not clear for me. Can you provide some comments? – pippp Apr 24 '16 at 23:40