I'm considering the following optimization problem:
$$\min_{\mathbf{A}\succ \mathbf{0}} \frac{\mbox{tr} \left(\mathbf{S}\mathbf{A}\right)}{\mbox{tr}\left(\mathbf{D}\mathbf{A}^{-1}\right)}$$
where matrix $\mathbf{A} \in \mathbb R^{d \times d}$ is symmetric and positive definite (SPD), matrix $\mathbf{A}^{-1}$ is the inverse of $\mathbf{A}$. Both $\mathbf{S} \in \mathbb R^{d \times d}$ and $\mathbf{D} \in \mathbb R^{d \times d}$ are SPD matrices. Does anyone know any solution to this problem?