Let us consider the following convex optimization problem:
$$\min_{\boldsymbol{\beta} \in \mathbb{R}^p} ||\mathbf{y} - \mathbf{X} \boldsymbol{\beta}||_2^2 + \sum_{g \in \mathcal{G}} \lambda_g ||\boldsymbol{\beta}_g||_2 + \sum_{g \in \mathcal{G}} \mu_g ||\boldsymbol{\beta}_g||_2^2,$$
where $\mathbf{y} \in \mathbb{R}^n$ is a known vector, $\mathbf{X} \in \mathbb{R}^{n\times p}$ is a known matrix, $\lambda_g$ and $\mu_g$ are non-negative known constants and $\mathcal{G}$ is a partition of the set $\{1, \ldots, p\}$.
If only the two first terms are considered, we obtain the objective function of the group lasso (for an appropriate choice of the constants $\lambda_g$). The third term acts by shrinking the coefficients to zero. I was wondering if the previous problem can be reformulated as
$$\min_{\boldsymbol{\beta} \in \mathbb{R}^p} ||\mathbf{y} - \mathbf{X} \boldsymbol{\beta}||_2^2 + \alpha\sum_{g \in \mathcal{G}} (\widetilde{\lambda}_g + \widetilde{\mu}_g)||\boldsymbol{\beta}_g||_2$$
for appropriate (and known) values of $\widetilde{\lambda}_g$ and $\widetilde{\mu}_g$, $g\in \mathcal{G}$, so that the efficient methods developed for solving the group lasso can be applied. The only reference to this topic that I found was https://stats.stackexchange.com/questions/296581/group-elastic-net, but I am not fully understand what is going on here.
Any help would be appreciated.