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Given

  • an arbitrary $(N \times N)$ square matrix ${\bf X}$
  • a positive definite $(M\times M)$ matrix ${\bf T}$
  • a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is exactly a single non-zero element in each row
  • a $(Q\times N^2)$ matrix ${\bf D}$ where there is exactly one non-zero element in each row and each column has at least one non-zero element

Define the function $$f({\bf X}) = vec({\bf X})^{\ast}{\bf D}^{\ast}({\bf Z}[{\bf T} \otimes {\bf X}({\bf I}+{\bf X}^{\ast}{\bf X})^{-1}{\bf X}^{\ast}]{\bf Z}^{\ast})^{-1} {\bf D}vec({\bf X})$$ where $\otimes$ denotes the Kronecker product. "$^{\ast}$" is Hermitian transpose.

Is $f({\bf X})$ quasi-convex? From simulations, this seems to be accurate. I checked for convexity, but it is not true (by numerical counterexamples)

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Can you check that definition again? There seems to be something wrong with your parentheses in that "$(I+X*X]$" doesn't look right. Also, while this doesn't sound from your description like a homework problem, MathOverflow still prefers questions ("Is $f(\mathbf{X})$ quasi-convex?") to commands ("Prove that $f(\mathbf{X})$ is quasi-convex."). –  Noah Stein Jun 13 at 8:55
    
Corrected. Sorry for the typos –  Michael Wallace Jun 13 at 9:13

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