Convexity proof of log-determinant $\log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right|$ with respect to $\bf X$

Question

Assume that ${\bf A} \in {\mathbb C}^{N\times N}$ is a positive semi-definite matrix, and $\bf B \in {\mathbb C}^{N\times N}$ is a positive definite matrix, i.e., ${\bf A} \succeq {\bf 0}$ and ${\bf B} \succ {\bf 0}$. ${\bf I}_N$ stands for the $N \times N$ dimensional identity matrix. ${\bf X} \in {\mathbb C}^{N\times N}$ is a matrix variabe and satisfies ${\bf X} \succeq {\bf 0}$. I think function \begin{equation} f({\bf X}) = \log\left|{\bf A}\left({\bf G}{\bf X}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}_N\right| \end{equation} is convex with respect to matrix $\bf X$. For the simple scalar case, \begin{equation} f(x) = \log\left(\frac{a}{|g|^2x + b} + 1\right), \end{equation} where $a \geq 0$, $b > 0$. It can be readily verified that $f(x)$ is convex. I tried to prove the convexity of $f({\bf X})$ by following similar steps in which Boyd proved the concavity of $\log |{\bf X}|$ (as attached), but failed. Could anyone provide some relevant hints or references? I would be very appreciated.

By the way, we can simplify $f({\bf X})$ as follows \begin{align} f({\bf X}) & = \log\left|{\bf G}{\bf X}{\bf G}^H+{\bf A} + {\bf B}\right| - \log\left|{\bf G}{\bf X}{\bf G}^H + {\bf B}\right|\\ & = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| + \log\left|{\bf A} + {\bf B}\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right| - \log\left|{\bf B}\right|. \end{align} So we only need to prove the convexity of \begin{equation} {\hat f}({\bf X}) = \log\left|{\bf G}^H\left({\bf A} + {\bf B}\right)^{-1}{\bf G}{\bf X}+ {\bf I}_N\right| - \log\left|{\bf G}^H{\bf B}^{-1}{\bf G}{\bf X} + {\bf I}_N\right|. \end{equation}

Answer 1

It has been proven in [1] that for given ${\bf A} \succeq {\bf 0},$ $h ({\bf X}) = \log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$ is convex over the positive semidefinite cone. We then show that $h ({\bf X})$ is a monotone nonincreasing function, i.e., for any ${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$, $h ({\bf X}_1) \leq h ({\bf X}_2)$. If ${\bf X}_1 \succeq {\bf X}_2 \succeq {\bf 0}$, we know that ${\bf A} {\bf X}_1^{-1} + {\bf I}_N \preceq {\bf A} {\bf X}_2^{-1} + {\bf I}_N$. Due to the monotonicity of $\log \det(\cdot)$, we have $h ({\bf X}_1) \leq h ({\bf X}_2)$. Hence, $h ({\bf X})$ is a monotone nonincreasing function.

Let $g({\bf X}) = {\bf G} {\bf X} {\bf G}^H + {\bf B} $, which is an affine function, and $f({\bf X}) = h(g({\bf X}))$. Then, from the condition which guarantees convexity of a composite function, [2, Subsection 3.2.4], it is known that $f({\bf X})$ is convex over the positive semidefinite cone.

[1]. Kim, KK.K., ''Optimization and Convexity of $\log \det({\bf A} {\bf X}^{-1} + {\bf I}_N)$'', Int. J. Control Autom. Syst., pp. 1067–1070, 2019.

[2]. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge university press, 2004.