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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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    $\begingroup$ Since you do not have convexity you have to use heuristics such as Genetic Algorithms. @enrico-schumann has written some code and some examples of these heuristics applied to Portfolio Optimization papers.ssrn.com/sol3/papers.cfm?abstract_id=3391756 $\endgroup$ – noob2 Oct 30 '20 at 9:24
  • $\begingroup$ ok thanks, I will have a look at it, I hope to find the answers to my 2 questions there. $\endgroup$ – Luigi87 Oct 30 '20 at 9:33
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    $\begingroup$ the following paper has mean-variance-skewness as a multi-objective function that they also convert to a single objective function, but then they turn to a GA model cscanada.net/index.php/mse/article/download/9233/10133 $\endgroup$ – develarist Nov 2 '20 at 8:40
  • $\begingroup$ thanks, I also found many here...researchgate.net/publication/… but I requested the full text and I need to wait $\endgroup$ – Luigi87 Nov 2 '20 at 8:49
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    $\begingroup$ i could send you Saranya Prasanna 2014. they use polynomial goal programming (PGP) to solve a mean-variance-skewness-kurtosis multi-objective optimization function (MVSK). nothing about heuristic or genetic algorithm. only cites Lai 1991 as being the first application of PGP to portfolio skewness as a work-around to the non-convexity $\endgroup$ – develarist Nov 3 '20 at 10:13

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