# Can a convex QCQP with an additional linear constraint be converted into a SOCP?

I have a quadratically constrained quadratic programming problem that I massaged into the form \begin{aligned} & \underset{x}{\text{minimize}} & & x^T Q x \\ & \text{subject to} & & x^T P x < \sigma^2 \\ &&& 0 \leq x_i \leq x_i^* \\ &&& Ax=b \end{aligned} where $Q$ and $P$ are positive semi-definite, $\sigma \geq 0$ and $x_i^* \geq 0 \, \forall i$.

Is this problem still convertible to a second-order cone problem so that I can use efficient solvers even with the linear constraint $Ax=b$? If so, what new variables would I need to add to the minimization to convert the problem?

It is always possible to convert a convex QCQP into an SOCP. It is not always possible to go the other direction, however. Just express this problem as follows: \begin{array}{ll} \text{minimize} & y \\ \text{subject to} & \|F x\|_2 \leq y \\ & \|G x\|_2 \leq \sigma \\ & 0 \preceq x \preceq x_i^* \\ & A x = b \end{array} where $F$ and $G$ are right square roots of $Q$ and $P$, respectively; that is, $F^TF=Q$ and $G^TG=P$. It really doesn't matter how you compute them; a Cholesky or will do, or a symmetric square root will do.