I have a large sparse quadratic optimization problem with a single quadratic constraint:
$$\begin{array}{ll} \text{maximize} & c'x\\ \text{subject to} & l \leq Ax \leq u\\ & x'Qx + b'x \leq u_q\end{array}$$
where
$x = (x_1, x_2, \dots, x_n)'$.
$A$ is a $k \times n$ matrix of linear constraint coefficients.
$l$ and $u$ are $k \times 1$ vectors of lower and upper bounds, respectively.
$Q$ is a $n \times n$ positive definite matrix.
$b$ is a $n \times 1$ vector.
$u_q$ is a scalar.
The solver I am using can handle quadratic objective function but only linear constraints. I was wondering if there is a way to reformulate the problem as:
$$\begin{array}{ll} \text{minimize} & \hat c'x + x'\hat Qx\\ \text{subject to} & \hat l \leq \hat Ax \leq \hat u\\\end{array}$$