\begin{array}{ll} \text{minimize} & \mbox{tr} (\mathrm A)\\ \text{subject to} & \mathrm A - \mathrm N \succeq \mathrm O_n\end{array} where $A$ and $N$ are pd matrices, and $A$ is diagonal.
There is a related post: Minimize trace of $A$ given that $A-N$ is positive semi-definite. . However, in that case $A$ is not diagonal thus, $tr(A)=tr(N)$ is possible, while in current case not.
For $A\in \mathbb{R}^{2\times2}$, I believe $\min tr(A)=\sum n_{ij}$, however for $A\in \mathbb{R}^{3\times3}$ we have inequality $\min tr(A)\leq\sum n_{ij}$. Can you please help with analytical approach so solve it