Optimization with positive definite constraint

Question

I am trying to solve for a given $Z$ and $\mu, \lambda > 0$: $$d(Z) = \min_{X \succ 0, Y} \left\{ - \log \det(X) + \langle S, X \rangle + \lambda \sum_{i \neq j} |Y_{ij}| + \frac{\mu}{2} ||Y||_F^2 + \langle Z, X-Y \rangle \right\}$$ which can be separated into $$ \min_{X \succ 0} \left\{ - \log \det(X) + \langle S + Z, X \rangle \right\} + \min_{Y} \left\{ \lambda \sum_{i \neq j} |Y_{ij}| + \frac{\mu}{2} ||Y||_F^2 - \langle Z, Y \rangle \right\}$$ $\DeclareMathOperator{\argmin}{argmin} \DeclareMathOperator{\prox}{prox} \DeclareMathOperator{\Shrink}{Shrink}$ For the second term, I think we can solve it by $$Y^* = \argmin_Y \left\{\lambda \sum_{i \neq j} |Y_{ij}| + \frac{\mu}{2} ||Y-Z||_F^2\right\} = \prox_{\lambda \sum_{i \neq j} |Z_{ij}|} (Z) = \Shrink(Z, \frac{\lambda}{\mu})$$ where Shrink is the off-diagonal shrink function. Is this the correct, since I am not sure about $\frac{\lambda}{\mu}$ constant. And I'm not sure how solve deal with the $X \succ 0$ constraint. Any help would be much appreciated!