Cubic and non convex optimisation problems in portfolio optimisation

Question

I have the following cost function for portfolio allocation, which considers also the skweness,

w are the weights to be found, mu is the return vector, Sigma is the covariance matrix, K is the co-skewness matrix, gamma is the risk aversion (let's say is a constant number).

This problem is not convex as far as I know and not quadratic but cubic..so how can I minimise it, in order to fall the global minimum rather that a local one?

I guess I cannot use cvxpy, what are the alternatives for such example or ingeneral for higher order moments?

Answer 1

I don't think this is convex either. That would imply you need a global MINLP solver (such as Baron, Couenne or Antigone). An alternative is to reformulate the problem as a non-convex quadratic problem (some extra variables and constraints are needed for this). This would allow you to use Gurobi.