I have the following portfolio cost function to maximise:
$$ w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w), $$
which considers the co-skewness ($M_3$ tensor), $γ$ is the risk aversion (a constant), $w$ is the weigh vector which is the quantity to estimate, $\Sigma$ is the covariance and $\mu$ the returns.
Now, to understand whether this function is convex or not and choose the best optimiser, I should compute the hessian. However, I have plenty of constraints, roughly 20 on my asset weighs, so even if potentially computing the hessian of that function can be done, unfortunately by adding all those constraints will change very much the optimisation hypersurface which I guess is almost impossibile to verify convexity.
So in case I have no idea whether a cost function is convex or not, is the genetic algorithm the only choice? What are its pros and cons for portfolio optmimisation?
Thanks. Luigi