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Is there a closed-form solution for $$\max_D \operatorname{Tr}(ADBD)$$ where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...

I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example. Let $...