Given symmetric positive semidefinite diagonal rank-$r$ matrix $R \in \mathbb{C}^{m \times m}$, where $r < m$, and scalar $p \geq 0$, I have the following optimization problem in matrix $X \in \mathbb{C}^{n \times m}$.
$A^{+}$ = pseudo inverse of A $$\begin{array}{ll} \underset{X \in \mathbb{C}^{n \times m}}{\text{minimize}} & \text{trace} \left( \left(R^{+} + \frac{X^{H}X}{\sigma^{2}} \right)^{+} \right)\\ \text{subject to} & \text{trace}\left( X^{H} X \right) \leq p\end{array}$$