The standard form for a Second Order Cone Program (SOCP)
\begin{equation} \begin{array}{c} \min _{x} f^{T} x \\ \left\|A_{i} x+b_{i}\right\|_{2} \leq c_{i}^{T} x+d_{i}, i=1, \ldots, m \end{array} \end{equation}
where $ A_{i} \in \mathbb{R}^{k_{i} \times n}, b_{i} \in \mathbb{R}^{k_{i}}, c_{i} \in \mathbb{R}^{n} \text { and } d_{i} \in \mathbb{R}$.
If the objective is quadratic instead we have
\begin{equation} \begin{array}{c} \min _{x} x^{T}\Sigma x \\ \left\|A_{i} x+b_{i}\right\|_{2} \leq c_{i}^{T} x+d_{i}, i=1, \ldots, m \end{array} \end{equation}
is there a classification for this type of problem? It looks like it is still a convex problem since both constraints and objective are convex.
Also, is there any way to proceed to solve this type of problem through KKT or any designed numerical solver to handle this problems?