# How do I reformulate this max GMV ratio constraint in convex way?

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!

• 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.
– Attack68
Commented May 17 at 19:15
• 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?
– inf
Commented May 17 at 19:50

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)