I read a paper about latent variable graphical models. Latent Variable Graphical Model Selection Via Convex Optimization. As stated in this title, $$ \begin{gathered} \left(\hat{S}_n, \hat{L}_n\right)=\underset{S, L}{\arg \min }-\ell\left(S-L ; \Sigma_O^n\right)+\lambda_n\left(\gamma\|S\|_1+\operatorname{tr}(L)\right) \\ \text { s.t. } S-L \succ 0, L \succeq 0 \end{gathered} $$ is a convex optimization problem. The log-likelihood function $\ell\left(K ; \Sigma_O^n\right) $ is $$ \ell\left(K ; \Sigma^n\right)=\log \operatorname{det}(K)-\operatorname{tr}\left(K \Sigma^n\right). $$ I know the graphical lasso is convex，but in the above optimization model, there are two matrix variables that need to be optimized. I have no idea to prove the convexity.
Thanks a lot.