I am recently faced with the following problem
\begin{array}\ \begin{align} \mathop {\min }\limits_{\bf F} \quad & \log\left|{\bf A}\left({\bf G}{\bf F}{\bf G}^H+{\bf B}\right)^{-1}+{\bf I}\right| \tag{1.a}\\ \text{s.t.} \quad & {\text {tr}}({\bf F}) \leq P, \tag{1.b}\\ & {\bf F} \succeq {\bf 0}, \tag{1.c} \end{align} \end{array} where $\bf A$ is a positive semi-definite matrix, $\bf B$ is a positive definite matrix, $P>0$, and all matrices are $N \times N$ dimensional square. This is a convex problem (please refer to another question for the convexity proof 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 $\bf X$). Hence, I hope to find the closed-form solution to $\bf F$ or at least find a way which can obtain the optimal solution quickly.
To solve problem (1), I referred to the water-filling method, which is usually applied to solve the following problem \begin{array}\ \begin{align} \mathop {\max }\limits_{\bf F} \quad & \log\left|{\bf G}{\bf F}{\bf G}^H+{\bf I}\right| \tag{2.a}\\ \text{s.t.} \quad & {\text {tr}}({\bf F}) \leq P, \tag{2.b}\\ & {\bf F} \succeq {\bf 0}. \tag{2.c} \end{align} \end{array} Denote the singular value decomposition of $\bf G$ by ${\bf W} {\bf V} {\bf T}^H$. Objective function (2.a) can then be written as \begin{align} \log\left|{\bf G}{\bf F}{\bf G}^H+{\bf I}\right| & = \log\left|{\bf W} {\bf V} {\bf T}^H {\bf F}{\bf T} {\bf V}^H {\bf W}^H+{\bf I}\right|\\ & =\log\left| {\bf V} {\hat {\bf F}} {\bf V}^H +{\bf I}\right|,\tag{3} \end{align} where ${\hat {\bf F}} = {\bf T}^H {\bf F}{\bf T}$. Problem (2) can then be solved by using water-filling.
However, this method does not work in solving problem (1). In particular, using eigenvalue decomposition, we can get ${\bf A} + {\bf B} = {\bf U}_1 {\bf D}_1 {\bf U}_1^H$ and ${\bf B} = {\bf U}_2 {\bf D}_2 {\bf U}_2^H$. Problem (1) is then equivalent to minimizing \begin{align} & \log\left|{\bf G}{\bf F}{\bf G}^H+{\bf A} + {\bf B}\right|- \log\left|{\bf A} + {\bf B}\right| -\log\left|{\bf G}{\bf F}{\bf G}^H + {\bf B}\right| + \log\left| {\bf B}\right|\\ =& \log\left|{\bar{\bf G}}{\bf F}{\bar{\bf G}}^H + {\bf I}\right|-\log\left|{\hat{\bf G}}{\bf F}{\hat{\bf G}}^H + {\bf I}\right|, \tag{4} \end{align} where ${\bar{\bf G}} = {\bf D}_1^{-\frac{1}{2}} {\bf U}_1^H {\bf G}$ and ${\hat{\bf G}} = {\bf D}_2^{-\frac{1}{2}} {\bf U}_2^H {\bf G}$. Denote the singular value decomposition of ${\bar{\bf G}}$ and ${\hat{\bf G}}$ by ${\bf W}_1 {\bf V}_1 {\bf T}_1^H$ and ${\bf W}_2 {\bf V}_2 {\bf T}_2^H$, respectively. Since ${\bf T}_1 \neq {\bf T}_2$, we cannot get a ${\hat {\bf F}}$ as in (3).
I would be very appreciated if someone could provide some relevant hints or references to solve problem (1).