I am trying to reformulate an existing SDP problem by changing the objective slightly. The problem must be an SDP, which means the constraints can be linear, SD, or SOCs. However, I need to add the following quadratic constraints, which I am trying to reformulate into SOCs:
$$ x_i \cdot(Q\ x)_i \leq z_i \quad \forall i$$
where $x,z \in \mathbb{R}^n$, $Q \in \mathbb{R}^{n \times n}$ is PSD, and the notation $(Q\ x)_i$ indicates the $i^{th}$ element of the vector $Q\ x$. I have been trying to think of ways to convert this into a computationally tractable format, but I am more familiar with traditional SOCP formulations such as going from $x^T Q\ x \leq t^2$ to $||Q^{1/2} x|| \leq t$, where $t \in \mathbb{R}^1$.
Does anyone know if it is even possible to transform this type of quadratic constraint into a SOC?
Take $Q = \begin{pmatrix}1&1\\1&1\end{pmatrix}$, which clearly is PSD. Now, look at the constraint $$ x_1 (Q x)_1 \leq 1 $$ which, by definition, is $$ x_1^2 + x_1 x_2 \leq z $$ The constraint above defines a non-convex set (set $z=0$ and plot it). Thus, the constraints you have are non-convex in general, and thus cannot be formulated as SOC or even semidefinite constraints. You might be lucky with a specific $Q$ which generates convex constraints.