How to solve a Semidefinite problem with fractional Objective function

Question

I have a Semidefinite problem of the form \begin{align} \min_{t\epsilon R,A \epsilon S_{+}^n}~& 1/t \\ &t>0\\ &A>=t C\\ &A\odot C_M=T_M\\ &\left\lvert\lvert A \right\rvert\rvert _F^2 <=r \end{align} or a more general form \begin{align} \min_{t\epsilon R,A \epsilon S_{+}^n}~& 1/t+b t \\ &t>0\\ &A>=t C\\ &A\odot C_M=T_M\\ &\left\lvert\lvert A \right\rvert\rvert _F^2 <=r \end{align} Which matrices $C_M , T_M$ have many zeros and $ C >=0$ is a constant matrix. How to solve this without adding any other Semidefinite constraint to the problem?(Because it makes solving it slover). How to solve it efficiently?