Non-convex QCQP Consider the following optimization problem:
$$\begin{array}{ll} \text{minimize} & \mathbf{x}^{T} \mathbf{A} \mathbf{x}\\
\text{subject to } & \mathbf{x}^{T} \mathbf{P}_i \mathbf{x} > 0, \quad  i \in \{1, \dots, n-1\}\\ & \mathbf{x}^{T} \mathbf{e}_1 = 1\end{array}
$$
where $\mathbf{x} \in \mathbb{R}^{n \times 1}$, $\mathbf{A} \in \mathbb{R}^{n \times n}$ is positive semidefinite.  The matrix $\mathbf{P}_i \in \mathbb{R}^{n \times n}$ is an all-zero matrix except $1$ at its $i^{\text{th}}$ diagonal entry and $-1$ at its $(i+1)^{\text{th}}$ diagonal entry. The vector $\mathbf{e}_1$ is the first column of the identity matrix.
It turns out that this problem is a non-convex QCQP since $\mathbf{P}_i$'s are not definite. 
This problem is to minimise a quadratic function, where $\mathbf{x}_1 = 1$ and $1 > \vert \mathbf{x}_2 \vert > \ldots > \vert \mathbf{x}_n \vert$. 
Any suggestions on how to solve this problem are highly appreciated. 
Thanks a lot.
 A: I would cast this as a mixed-integer quadratic program. To do so, we first define new continuous variables $y_2,\dots,y_n$ and introduce these constraints:
$$-y_k\leq x_k\leq y_k, \quad k=2,3,\dots, n, \quad y_2=1$$
Then we define new binary variables $z_2,\dots,z_{n-1}\in\{0,1\}$:
$$x_k \geq y_{k+1} - 2 z_k, \quad -x_k \geq y_{k+1} - 2 ( 1 - z_k ), \quad k=2,3,\dots, n-1$$
This implies that, for any feasible solution,
$$ x_k \geq y_{k+1} \geq |x_{k+1}| \quad \text{or} \quad
   - x_k \geq y_{k+1} \geq |x_{k+1}|
\quad\Longrightarrow\quad |x_k| \geq |x_{k+1}|$$
These additional variables and constraints replace the quadratic constraints, leaving you with this:
\begin{array}{ll}
\text{minimize}   & \sum_{ij} A_{ij} x_ix_j \\
\text{subject to} & -y_k \leq x_k \leq y_k, \quad k=2,3,\dots, n \\
                  & x_k + 2 z_k \geq y_{k+1}, \quad k=2,3,\dots, n-1 \\
                  & -x_k + 2(1-z_k) \geq y_{k+1}, \quad k=2,3,\dots, n-1 \\
                  & x_1 = y_2 = 1 \\
                  & z_2, z_3, \dots, z_{n-1} \in\{0,1\}
\end{array}
This can be solved with any mixed-integer quadratic programming solver.
