Including optimization constraint in objective function

Question

Does solving the following optimization problem : $ \min_A \text{Tr} [A \sigma] $ such that $ \text{Tr} [A^{-1} \rho] \leq 1 $ give the same solution as solving $ \min_A \text{Tr} [A \sigma]\text{Tr} [A^{-1} \rho] $?, where $\rho, \sigma \geq 0 $ and $\text{Tr } \rho= \text{Tr } \sigma=1 $, as they are density matrices. Thank you!