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Assuming I have N stocks. I want to have the following constraint in my optimization problem setup.

$|x_i| \le \alpha \sum_{j}^N |x_j|$ where $\alpha$ is known, say 0.6. The intuition here is the GMV of every optimized stock can't be more than $\alpha$ of the optimized overall GMV, to achieve a more diversified portfolio.

Assuming N = 3, then we have:

$|x_0| \le 0.6 (|x_0| + |x_1| + |x_2|)$

$|x_1| \le 0.6 (|x_0| + |x_1| + |x_2|)$

$|x_2| \le 0.6 (|x_0| + |x_1| + |x_2|)$

How do I re-formulate this in a convex way? The solve I'm using is Mosek.

Thanks!

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  • $\begingroup$ This isn't technically an answer, because I can't immediately observe if what you are trying to do is achieveable but, since $\alpha$ seems somewhat subjective in your query would you consider readjusting this subjectivity: $ \alpha_1 \sum_i |x_i| \approx \alpha_2 ||\mathbf{x}||_2 $ Then you solve this with SOCP. $\endgroup$
    – Attack68
    Commented May 17 at 19:15
  • $\begingroup$ Thanks. Can you elaborate on how to solve $\alpha_1 \sum_i |x_i| \approx \alpha_2 || x||_{2}$ in SOCP? Following the thoughts, what about if the constraint is changed to: $x_i^2 \le \alpha \sum_{j}^N x_j^2$ Is there a way to reformulate this one? $\endgroup$
    – inf
    Commented May 17 at 19:50

1 Answer 1

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The closest I have come so far is to engineer a the constraint into a quadratically constrained quadratic program (QCQP):

enter image description here

where in your case, $q_i=0$ and $r_i=0$ and, $ P_i = \mathbf{I} - \alpha \mathbf{1} $ (where $\mathbf{1}$ is a matrix of ones)

I am assuming that your objective function is probably quaudratic as well according to most portfolio optimization problems.

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