Choosing Optimizers for Portfolio Objectives with Co-Skewness

Summary

The document presents a portfolio objective combining expected return, a covariance-based risk penalty, and a third-moment co-skewness term. It asks how to choose an optimizer when the objective’s convexity is uncertain and many weight constraints may shape the feasible region. The objective captures a preference that depends on more than mean and variance, while its higher-order term can complicate optimization.

The discussion raises genetic algorithms as a possible approach but provides no answer, experiments, or comparison with other methods. It therefore offers a useful problem formulation and identifies convexity and constraints as practical concerns, but it does not establish that a genetic algorithm is necessary or suitable. The document gives no evidence about convergence, solution quality, or computational cost, so those tradeoffs remain unresolved.

Key ideas

• The objective combines expected return, covariance risk, and a co-skewness contribution.
• A third-moment term can make optimization more complex than a mean–variance problem.
• Numerous weight constraints may affect the feasible region and complicate convexity analysis.
• The document asks about genetic algorithms but supplies no optimizer comparison or empirical evidence.

Tags

# Can genetic algorithm help in portfolio optimisation when convexity is not verifiable


# Can genetic algorithm help in portfolio optimisation when convexity is not verifiable












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

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