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I have a $N\times N$ symmetric positive semidefinite matrix $Q$, and am considering a class of symmetric positive definite matrices having all eigenvalues in a given bounded interval $[a, b]$.

Is it possible to find a matrix in this class such that the sum of the smallest $k$ elements of $\text{diag}(XQ)$ is minimized?

I am welcoming any suggestion, either in the form "not a trivial problem", or "here is the solution, or a pointer to it".

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