Hi I have the following problem whose solution has lured me for some months now....

All matrices are complex $N\times N$. Let $A$ be a positive definite matrix with all eigenvalues strictly smaller than one. Let $0<\rho < 1$. I need to maximize, preferably in closed form or a "recipe" how to get the optimal solution, the following function

$$\max_{X,Y}f(X,Y) = \log \det (X) + \mathcal{R}\left(\mathrm{Tr}\left((A-I)(X+2\rho Y)\right)\right) + \mathrm{Tr}\left((\rho^2 A-\rho I)Y^HX^{-1}Y\right)$$ where $X$ is a positive definite matrix that only takes non-zero values along the center $2K+1$ main diagonals and $Y$ is an arbitrary matrix that must have zeros along the center $2K+1$ main diagonals (i.e., $Y$ is "alive" where $X$ is not and vice versa).

I am interested in all values of $K$ (from 0 to $N-1$, but the two extreme cases are simpler).

There is no further structure in $A$, it can any matrix with the above described properties. Note that the optimization over $Y$ alone is simple as the expression is quadratic in $Y$, but the remaining optimization over $X$ is then difficult. The optimization over $X$ alone seems to be convex.

$\mathcal{R}(\cdot)$ means "real part of" and $\mathrm{Tr}(\cdot)$ means "Trace".

Any help would be greatly appreciated (perhaps even a joint paper). Even results in the form of structure properties would be of interest.