optimization with absolute constraints

Question

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?

Answer 1

Why not just do:

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

Google for LP absolute value constraint transformations. Here is a helpful online tutorial.

And if these are portfolio weights, don't forget that they should add up to 1.