Here we have this optimization problem:
Given positive semi-definite matrix $A \in \mathbb{R}^{n \times n }$, and matrix $B \in \mathbb{R}^{n \times m} \text{ and vectors } d \in \mathbb{R}^n,e \in \mathbb{R}^ m$, we seek to find $x$ be binary variable of length $n$
\begin{equation} \label{eq1} \begin{split} & \min_x x^T Ax + x^Td \\ & \text{subject to } B^Tx = e \end{split} \end{equation} However, $n$ can be as large as 1000. I tried cvxpy with backend sovler ECOS_bb, but it was too slow to solve.
Is there any solver that I could use to solve this problem ?
Should I linearize the objective $A$ to convert it into a mixed integer linear programming problem ?
Thanks for any advice and help !!