# Portfolio optimization with non-linear cost

I am trying to solve a mean-variance problem with a non-linear market impact cost term in there. This is the problem I am trying to solve

$$\max_x \left ( \alpha x - \gamma x' \Sigma x - a\sqrt{|x-x_0|} \right ) \quad s.t. \quad \text{unconstrained}.$$

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.

• @MatthewGunn .. I understand that, but the square root function is the most widely used market impact cost model. I was just trying to figure out how practitioners are able to incorporate such a model in a portfolio optimization framework. Maybe they use approximations like a piece-wise linear function ? – silencer Jul 30 '18 at 17:20
• What makes an optimization problem easy vs. hard isn't linear vs. non-linear. It's convex vs. non-convex. A starting point may be to use fminunc or fmincon and see if you can get reasonable results. If you have few variables, you can grid search. – Matthew Gunn Jul 31 '18 at 12:27

You're not going to be able to solve it with quadprog because $-x^2 - \sqrt{|x|}$ can't be represented as a quadratic function. While the original problem, $\max_\mathbf{x} \mathbf{a} \cdot \mathbf{x} - \gamma \mathbf{x}'\Sigma \mathbf{x}$ is a convex optimization problem (and convex problems can, in general, be efficiently solved), your problem isn't convex. Your objective for maximization is a sum of a concave and a quasi-convex. You can try using fmincon, but be aware it may give you local rather than global optima. There's reasonable stuff you can do, but it's not a trivial problem to find a global solution.
Notice the local optima. 