I am trying to solve a mean-variance (quadratic optimization) problem with a non-linear market impact cost term in there. This is the problem I am trying to solve
$$\max \alpha a - \gamma x^T \Sigma x - a\sqrt{|x-x_0|} $$
where, $x_0$ is the current portfolio holding.
I am using MATLAB's quadprog program to solve this problem. How can I setup my optimization problem correctly to incorporate the sqrt term in the objective ? I am sure there is a way to do this, but I am not aware of it.