How to transform a cubic optimisation problem into a quadratic for portfolio allocation

Question

I have the following cost function for portfolio allocation:

$$ w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w), $$

which considers also the co-skewness ($M_3$ tensor), $\gamma$ is the risk aversion (a constant)

This function is cubic and non 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 non-convex but now quadratic form, which can be solved probably more easily.

Can anyone help please to reformulate the above equation in order to make it quadratic? Can I still use cvxpy for non-convex optimisation?

This question is a follow-up to:

• Is quadratic programming used to maximize portfolio skewness and kurtosis?
• How to add the effect of skewness in the portfolio optimisation objective function?

Answer 1

I don't think you can reformulate the problem as written to be quadratic, but you can "cheat", and approximate it as a quadratic problem locally. That's the general idea behind using Newton's method in optimization. If

$$ f(w) := w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w) $$

is your optimization function, then, thanks to the Taylor series, at any $w_k \in \mathbb{R}$ with all the required derivatives defined,

$$ f(w_k) \approx f(w_k) + \nabla f(w_k)^Tw_k + \frac{1}{2}w_k^T\nabla^2f(w_k)w_k^T $$

in some neighborhood of $w_k$. Here $\nabla f(w_k)$ and $\nabla^2 f(w_k)$ are the gradient and Hessian matrix, respectively. Under some fairly general conditions, like boundedness near the minimum, or constraints on the size of the derivative, you can show that Newton's method will converge superlinearly to a minimum.

That solves one of your two problems, which is how to deal with cubic terms. You iteratively approximate your function as a quadratic at each step, until you find a minimum. In practice, this can work quite well for a wide class of problems.

The second problem you mentioned is that the function is non-convex, which poses some risks:

1. The function has local minimum, and your optimization method converges on a non-global, local minimum.
2. The function has no minimum at all, and your optimization method sputters on until it hits some maximum number of steps and gives up.

There are some techniques that can be used to deal with local minimum, but there's usually not a guarantee that they will find a global minimum. Stochastic optimization methods are built to deal with this type of problem, but you don't always need the complexity these introduce. In particular, if you have some domain knowledge you can use to find a "close" initial guess of the minimum, Newton's method should converge to the global minimum when it exists.