I have the following cost function for portfolio allocation to be maximise:
$$ w^\top \mu-\frac{1}{2}\gamma w^\top \Sigma w+\frac{1}{6}\gamma^2 w^\top M_3(w\otimes w), $$
which considers the co-skewness ($M_3$ tensor), $\gamma$ is the risk aversion (a constant), $w$ is the weight vector which is the quantity to estimate, $\Sigma$ is the covariance and $r$ the returns.
This function is cubic and non convex (I have not calculated the hessian but I think everytime the higher polynomial degree is 3 then is never convex..), so I cannot use the typical convex optimisation with cvxpy in python. However, it should be possible to transform/replace the cubic term with a quadratic term and adding a new constraint in order to have a quadratic form, which can be solved probably more easily. Obviously if the transformation gives a convex quadratic form is better, but if it does not, also quadratic non-convex is fine as there are solvers for non-convex quadratic problems (Gurobi).
Can anyone help please to reformulate the above equation in order to make it quadratic?
Thanks