# optimization with absolute constraints

Suppose I have an optimization where I need to impose ADV-like constraint (for a case where Shorting is allowed):

$\max \mu'w - \lambda w'\Sigma w$

$|w| \le V$

$Aw = 0$

and I want to use a Quadratic Programming formulation. I read somewhere that I can replace $|w| = z$ by two inequalities. Which one of the two is valid:

Case 1

$\max \mu'w - \lambda w'\Sigma w - M z$

$z \le V$

$w \le z$

$-w \le z$

$Aw = 0$

where $M$ is a very large constraint, which I think will force $|w| = z$

Case 2

$\max \mu'w - \lambda w'\Sigma w$

$z \le V$

$w \le z$

$-w \le z$

$Aw = 0$

Case 2 above is what I saw a few places on the net, but it got me thinking that this constraint is equivalent to $|w| \le z$ and I need to find another way to force the equality.

Is either Case 1 or Case 2 or both a correct way to handle the $|w| \le V$ constraint?

$$max \,\, \mu ^T w - \lambda w^T \Sigma w$$ s.t.: $$w \leq V$$ $$-w \leq V$$ $$A w = 0$$