Portfolio optimization

Question

first I just hope that this question is in the right place.

I have started working on portfolio optimization and the formulation of the problem and their solution : For example in the Markowitz problem lowest variance of the portfolio for a certain level of return can be express as for n risky assets

min $\ \frac{1}{2} \theta^TQ\theta$

s.t. $\ \mu^{T}\theta = \mu^{*}$

and
$\ \mathbb{1}^{T}\theta = 1$

where $\ \theta \in \mathbb{R}^{n} $ is the the weight of the risky asset in our portfolio $\ \mu \in \mathbb{R}^{n}$ is the expected return of the risky asset and $\ \mu^{*} \in \mathbb{R}^{+} $ is the objective function.

Now and it is my problem, I would like to formulate in terms of optimization problem the following problem/ let's say I have initial wealth X, I still want to determine the portfolio with the lowest variance for a certain expected return $\ \mu^{*} $, I allow short selling but I also want that the money raised by short selling stay under a certain level that depends on X for example that the level of money raised by short selling be inferior than X/5.

Thank you for any hints or suggestions, I hope I did not forget some hypothesis and I made myself clear.

Answer 1

The problem as you formulate it above already allows for short-selling. You just have to add the constraint: $$ \theta_i \ge l $$ where $l$ is the lower bound. This is equivalent to $$ -\theta_i \le -l $$ which if often the way linear constraints are formulated. Any solver that is able to work with box-constaints can solve this.