I use CVX to minimize objective function with sparse group lasso penalty. Y is [N x k], X is [N x p], B is the parameter matrix [p x k]. the set of independent variables have been broken down into G groups. I want to find the \hat{B} to minimize the following objective

Mathematical formulation:

\begin{eqnarray}

\underset{B}{Min}~ \bigg{\frac{1}{2n}| Y-\sum_{g\in G}X^{(g)}B_{(g)}|*F^2 +\lambda \big((1-a) \sum*{g \in G} \eta_g\lVert B_{(g)}\rVert_2 +a \sum_{i,j}| B_{i,j}|\big) \bigg}

\end{eqnarray}

I use CVX to get \hat{B}, but I am not sure about the algorithm and property of \hat{B}, how is the asymptotic properties like consistency and approximated normality of the estimates \hat{B}?