Trace minimization problem with "block diagonal diagonal" constraints?

I've reduced my optimization problem to the following trace minimization problem: $$\min_X\text{tr}(AXB),$$ subject to that $X$ is a block diagonal matrix whose blocks are all the same -- a diagonal matrix $\Sigma$: $$X=\begin{pmatrix} \Sigma & \bf0 & \cdots \\ \bf0 & \Sigma & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}.$$

Not sure if this helps, but each element of $\Sigma$ is guaranteed $>0$.

How should I impose this constraint on $X$?

P.S.: I'm sure it's stupid to call $X$ "block diagonal diagonal" -- please suggest a better name :-)

Partition $A$ and $B$ so that they have conforming block structures to $X$. Let $A_{ij}$ denotes the $(i,j)$-th sub-block and similarly for $B$. Then $$\operatorname{tr}(AXB)=\operatorname{tr}\sum_{i,k}(A_{ik}\Sigma B_{ki})=\operatorname{tr}\left(\left(\sum_{i,k}B_{ki}A_{ik}\right)\Sigma\right)=\langle\mathbf c,\mathbf s\rangle,$$ where $\mathbf c$ is the diagonal of $C=\sum_{i,k}B_{ki}A_{ik}$ and $\mathbf s$ is the diagonal of $\Sigma$. Now,
• $\inf_{\mathbf s>0}\,\langle\mathbf c,\mathbf s\rangle=-\infty$ when $\mathbf c$ has a negative entry,
• $\inf_{\mathbf s>0}\langle\mathbf c,\mathbf s\rangle=0$ when all elements of $\mathbf c$ are nonnegative; unless $\mathbf c$ has a zero entry, this infinum is not attainable.
• @SibbsGambling The term "minimum" means something attainable. In your case, since $\mathbf s$ is required to be entrywise positive, if $\mathbf c$ is also entrywise positive, then the dot product between the two vectors approaches zero when $\mathbf s$ tends to zero. So zero is an infinum of the dot product. Yet this infinum is not attainable, because the dot product of two entrywise positive vectors must be positive. Commented Jul 26, 2016 at 16:30