I'm dealing with the portfolio optimization case study of mosek and want to add limits on the total number of assets to be re-weighted and the turnover of each asset.
The objective is to maximize Sharpe ratio $\frac{\mu^{T}w}{\sqrt{w^T \Sigma w}}$ or to minimize $\frac{\sqrt{w^T \Sigma w}}{\mu^{T}w}$ with $\mu^T w \ge 0$ and budget constraint $e^Tw=1$. It's similar to linear fractional problem, so I introduce a variable $g$ to get rid of the denominator. $$ \min_{\hat{w}} \sqrt{\hat{w}^T \Sigma \hat{w}} $$ where $\hat{w}$ is the scaled portfolio weights that $\hat{w} = w * g$. By cholesky decomposition we can have $GG^T = \Sigma$ and $\sqrt{\hat{w}^T \Sigma \hat{w}}=\sqrt{\hat{w}^TGG^T\hat{w}}=||G^T\hat{w}||_2$ which can be expressed by a quadratic cone $(s, t) \in Q^{n+1}$ and $t = G^T\hat{w}$
$$ e^{T}\hat{w} = g $$
$$ \mu^{T}\hat{w} = 1 $$
$$ \hat{w}_{i} - \hat{w}_{0_{i}} \le \hat{z}_{i} $$ $$ \hat{w}_{0_{i}} - \hat{w}_{i} \le \hat{z}_{i} \quad (\hat{z}_{i}=|\hat{w}_i - \hat{w}_{0_i}|) $$ $$ \hat{z}_{i} \le g * U_{i}y_{i} $$ $$ e^{T}y \le k $$ $$ -\hat{w}_{i} \le 0 $$ $$ -g \le 0 $$
$$ y\in \{0, 1\} , y \in Z $$
$U_{i}$ is given as a limit on each asset. $z_i$ is the turnover of each asset, and $g$ is the re-scale factor. $y_{i}$ works as an indicator to limit the number of assets to be re-weighted.
So here we have $z_i \leq g * y_{i}$, while $g, y_{i}, z_{i} \geq 0$. I'm not sure whether it's convex or not. If so, how to deal with this inequality.
I tried to add another variable $c$ that $c_i^2 = z_i$ so we can have a rotated quadratic cone $g * y_i \ge c_i^2$. But I don't know how to link $c$ with $z$ either.