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I recently came across the following method for portfolio optimization: Let $Y$ be a random variable that describes the returns of $n$ assets. Fix a constraint matrix $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$. Then, we sample from $Y$ and calculate the weights $x$ that maximize returns under the constraints $Ax \ge b$. Finally, we average over all optimal allocations to compute the final weights $\bar x$.

Since the set $\{ Ax \ge b\} $ is convex, we can be sure that $\bar x$ also satisfies the constraints. Thus, I think the practical idea is that $\bar x$ will be a good feasible compromise between extreme cases of $Y$ while still somewhat maximizing the objective function.

But from a statistical point of view, it is unclear to be why this would be a good procedure. Why don't we use the mean of $Y$ directly? Is this some kind of Bayesian approach?

Update

I think my statistical concerns were essentially adressed here and the method was discussed in the paper mentioned there (though with a different objective function including a quadratic term that reflects risk)

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    $\begingroup$ Interesting. Where did you see ths algorithm? $\endgroup$
    – nbbo2
    Commented Apr 9, 2022 at 15:37
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    $\begingroup$ Such linear constraints will not reflect risk. So this will miss one of the two interesting aspects of portfolio optimization. $\endgroup$
    – g g
    Commented Apr 9, 2022 at 17:51
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    $\begingroup$ Hi Claudio: This is the paper that I was referring to. I don't remember if it's enough for learning how to go about dealing with a specific problem but I remember it being quite helpful for someone without previous exposure. I hope it helps some. www2.stat.duke.edu/~scs/Courses/Stat376/Papers/Basic/… $\endgroup$
    – mark leeds
    Commented Apr 10, 2022 at 10:55
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    $\begingroup$ The approach is also called Michaud portfolio resampling. en.wikipedia.org/wiki/Resampled_efficient_frontier $\endgroup$
    – John
    Commented Apr 11, 2022 at 15:11
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    $\begingroup$ The argument against resampling is typically 1) it generates less expected utility than a classical mean-variance optimal portfolio, 2) the smooth behavior of the resampled frontier is only produced when inequality constraints are used (most common being the long only) $\endgroup$
    – John
    Commented Apr 11, 2022 at 15:13

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