The following problem has come up in my studies of logarithmic norms.
I wish to find $\mu \in \mathbb{R}$ and a positive semidefinite $B$ so as to minimize the convex function $c \mu - \log\det(B)$ subject to $A \preceq B$ and $JB + BJ^T \preceq 2\mu B$ where $A$ is some positive semidefinite matrix, $J$ is a general matrix and $c$ is a positive constant.
This problem is nearly convex, but the product $\mu B$ in the second constraint is problematic. Is there any way to transform this to a convex problem?