A non-convex quadratically constrained quadratic program (QCQP) How can I solve the following quadratically constrained quadratic optimization problem?
$$\max_{x} \qquad x'Ax$$
such that
$$x'P_ix - k \leq 0 \qquad i=1,2$$
$A, P_i$ are positive semidefinite matrix. I know we can minimize $x' \hat{A} x$ where $\hat{A} = -A$ but this changes the problem to concave one with convex constraints. $x \in \mathbb{R^2}$. If it helps, finding the minimum of the inverse of $x' A x$ is also an option.
The objective function in that case would be $$\min_x \qquad \frac{1}{x'Ax}$$
 A: We have the following QCQP
$$\begin{array}{ll} \text{maximize} & \mathrm x^{\top} \mathrm A \, \mathrm x\\ \text{subject to} & \mathrm x^{\top} \mathrm P_1 \mathrm x \leq q_1\\ & \mathrm x^{\top} \mathrm P_2 \mathrm x \leq q_2\end{array}$$
where $\mathrm A, \mathrm P_1, \mathrm P_2 \in \mathbb R^{n \times n}$ are symmetric and positive semidefinite, and $q_1, q_2 > 0$. Note that we have a convex objective function and two convex inequality constraints. However, since we want to maximize the objective, the given QCQP is non-convex.
If $\mathrm P_1, \mathrm P_2$ were positive definite (thus, invertible), then, using the Schur complement, we would be able to write the inequality constraint $\mathrm x^{\top} \mathrm P_i \mathrm x \leq q_i$ as the following linear matrix inequality (LMI) 
$$\begin{bmatrix} \mathrm P_i^{-1} & \mathrm x\\ \mathrm x^{\top} & q_i\end{bmatrix} \succeq \mathrm O_{n+1}$$
We would then have the LMI-constrained quadratic optimization problem
$$\begin{array}{ll} \text{maximize} & \mathrm x^{\top} \mathrm A \, \mathrm x\\ \text{subject to} & \begin{bmatrix} \mathrm P_1^{-1} & \mathrm x & & \\ \mathrm x^{\top} & q_1 & & \\ & & \mathrm P_2^{-1} & \mathrm x \\ & & \mathrm x^{\top} & q_2\end{bmatrix} \succeq \mathrm O_{2n+2}\end{array}$$
Does anyone know of any work on quadratic optimization over (convex) spectrahedra? 
Take a look at Didier Henrion's lecture notes and the references therein.  
