Consistency of estimate using CVX

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:
\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}

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}?

This question is out of scope of this forum, and may be better suited for a statistically oriented forum such as

Thank you so much for your suggestion. I guess it will also help if we can know the algorithm for the objective minimization in CVX, so could you please give me some suggestions about where to find the algorithm of this procedure? Thank you very much in advance.

I don’t think knowing the algorithm used to find the minimum is particularly relevant to your question. CVX performs some transformations, then calls a solver to solve the problem, then transforms the solver results back to your original formulation. The exact algorithm used may depend on the solver employed.

Oh, yes, I made some misunderstanding about it. Thank you!