I am working on a binary optimization problem. So far I have derived the following constraint functions. \begin{align} \begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^T \right) A \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^T \right) & v \\ v^T & t \end{bmatrix} \succeq 0 \end{align} , where the optimization variables are $x_{i, j} \in \{0, 1\}$. $P$ and $A$ are given positive definite symmetric matrices, $v$ is a vector, and $\alpha$, $t$ are constants.

The quadratic matrix inequality seems to be non-convex in general, so cvx solvers cannot directly accept it. I am not sure whether it is still non-convex when the domain is real numbers between 0 and 1. I have attempted to solve the problem through semidefinite relaxation. However, since the problem is not homogeneous quadratic (there are linear terms), the solution quality turned out to be poor. I am looking for any optimization techniques that may help solving this problem. Thanks for any suggestion in advance.