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)
