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?