I have an optimization problem as follows.
$$ \text{maximize}_{\mathbf{x}\in \mathbb{R}^{n}} \sum_{i=1}^{P} \frac{(\mathbf{x}^{T} H_{p} \mathbf{x})}{(\mathbf{x}^{T} C_{p} \mathbf{x})}$$ $$s.t. A\mathbf{x}=b, F\mathbf{x} \le d, \textbf{x} \in \{0,1\}^{n}$$
The objective function is a sum of fractional functions of trace of quadratic form where $C_{p}$ is rank-1 PSD matrix. $H_{p}$ is a $n\times n$ size PSD matrix. All the variables and constants are on the real domain. $A$ and $F$ are $m_{1}\times n$ and $m_{2}\times n$ matrices, respectively. I would like to get any suggestion or approaches to solve this problem.
My approach to solve this problem was using the SDP relaxation with introducing a new variable $\mathbf{Y}=\mathbf{xx}^{T}$. Then we have
$$ \text{maximize}_{\mathbf{Z}\in \mathbb{R}^{n+1 \times n+1}} \sum_{i=1}^{P} \frac{\text{Tr}(\hat{H}_{p} \mathbf{Z})}{\text{Tr}(\hat{C}_{p} \mathbf{Z})}$$ $$s.t. \text{Tr}(\hat{A}_{j}\mathbf{Z})=b_{j}, \text{Tr}(\hat{F}_{k}\mathbf{Z}) \le d_{k}, \text{Tr}(\hat{I}_{l}\mathbf{Z})=0$$ $$ \mathbf{Z}_{n+1,n+1}=1, \mathbf{Z} \succcurlyeq 0.$$
I just simplify the constraints for the new variable $$ \mathbf{Z} = \begin{bmatrix} \mathbf{Y} & \mathbf{x} \\ \mathbf{x}^{T} & 1 \end{bmatrix}. $$
All the matrices with $\hat{\cdot}$ (hat) notation is a $(n+1 \times n+1)$ size real symmetric matrix corresponding to the objective function and the constraints. For example, $$ \hat{H}_{p} = \begin{bmatrix} H_{p} & 0 \\ 0 & 0 \end{bmatrix}, $$
$$ \hat{A}_{j} = \begin{bmatrix} 0 & a_{j}/2 \\ a^{T}_{j}/2 & 0 \end{bmatrix} $$ where $a_j$ is a $j$-th column of the matrix $A$ and $b_j$ is a $j$-th element of the vector $b$. We can do the same thing for the matrix $C_p$ and $F$. Please note that the above matrix expressions are block matrix expressions. The boolean constraint can be replaced by $\text{Tr}(\hat{I}_{l}\mathbf{Z})=0$, for example, $$ \hat{I}_{1} = \begin{bmatrix} 1 & 0 & \cdots & -1/2 \\ 0 & \ddots & \cdots & 0 \\ \vdots & \cdots & \cdots & \vdots \\ -1/2 & 0 & \cdots & 0 \\ \end{bmatrix}. $$
Now the problem has an objective function of a sum of linear fractional-functions. The first 4 constraints are linear constraints and finally, without the rank-1 constraint, $\mathbf{Z}$ should be a PSD matrix.
At this moment, even the relaxed version of the problem looks hard to solve directly. I would appreciate any help.