I have a QCQP as shown below: \begin{equation} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x}^{T}\cdot\mathbf{P}\cdot\mathbf{x}\\ & \text{subject to} & & \mathbf{x}^{T}\cdot\mathbf{P}_{i}\cdot\mathbf{x}-r_{eq}^{i}=0, & & i\in S_{eq}\\ & & & \mathbf{x}^{T}\cdot\mathbf{P}_{j}\cdot\mathbf{x}-r_{max}^{j}<0, & & j\in S_{iq}\\ & & & \mathbf{x}>\mathbf{x}_{min}\\ & & & \mathbf{x}<\mathbf{x}_{max} \end{aligned} \end{equation}
$\mathbf{P}$ is symmetric and positive semidefinite. $\mathbf{P}_i$ and $\mathbf{P}_j$ are symmetric but not PSD as they have the following particular shape:
$ \mathbf{P}_{i}=\begin{bmatrix}0 & \dots & 0\\ \vdots & \ddots & \vdots\\ p_{i1} & \dots & p_{in}\\ \vdots & \ddots & \vdots\\ 0 & \dots & 0 \end{bmatrix}+\begin{bmatrix}0 & \dots & 0\\ \vdots & \ddots & \vdots\\ p_{i1} & \dots & p_{in}\\ \vdots & \ddots & \vdots\\ 0 & \dots & 0 \end{bmatrix}^{T} $
My approach is to relax the constraints using a Taylor approximation and convert the problem to a convex one. This provides decent results, depending, obviously, on the point that I use for the Taylor approximation.
Recently I stumbled over https://web.stanford.edu/class/ee392o/relaxations.pdf and I tried to set the problem as a SDP as below:
\begin{aligned} & \underset{\mathbf{x,X}}{\text{minimize}} & & \mathbf{Tr(XP)}\\ & \text{subject to} & & \mathbf{Tr(X}\mathbf{P}_{i})-r_{eq}^{i}=0, & & i\in S_{eq}\\ & & & \mathbf{Tr(X}\mathbf{P}_{j})-r_{max}^{j}<0, & & j\in S_{iq}\\ & & & \left[\begin{array}{cc} \mathbf{X} & \mathbf{x}\\ \mathbf{x^{T}} & 1 \end{array}\right]\mathbf{\succeq0}\\ & & & \mathbf{x}>\mathbf{x}_{min}\\ & & & \mathbf{x}<\mathbf{x}_{max} \end{aligned}
However, from what I understand, the SDP formulation only provides a lower bound on the objective function. Then, in order to get an actual feasible point, one has to run an iterative process in which the problem is linearised and solved again and again. However, in my case, due to the equality constraints obtaining random feasible points is not that easy.
While digging on the subject I also found out about SOCP. However, I was not able to find any resources that show how to convert a non-convex QCQP to a SOCP. I found some examples for the cases where all the $\mathbf{P}$ matrices are PSD, but none that address problems similar to mine.
So my questions are as follows:
Can a nonconvex QCQP problem as the one described above be set in a SOCP form?
If yes, does the SOCP solver provides (as in the case of SDP) just a lower bound on the objective function or also a feasible point for the original problem?
Are there some good techniques for obtaining feasible starting points for the iterative process mentioned above (the one that runs after solving the SDP)?
Any other ideas on how could I approach the original problem?