Convex constraint in a Mixed-Integer Program

Question

I have an optimization with the following convex constraint:

\begin{equation*} x_1^2+x_2^2\leq \textrm{C}_1\cdot x_3 + \textrm{C}_2\\ \end{equation*}

My problem also contains some integer variables ($x_1$, $x_2$ and $x_3$ are however continuous).

I know that there are efficient solvers for Mixed-Integer Second Order Cone Programs. Could you help me understand if my constraint is conic quadratic representable, so that I could implement it in an MISOCP? If not, any advice on how I could implement this constraint in a Mixed-Integer Program that can be solved efficiently with available solvers?

Answer 1

Yes, from the generic $x^TR^TRx \leq c^Tc+b$ being SOCP representable as $\left|\left|\begin{matrix}2Rx\\1-(c^Tx+b)\end{matrix}\right|\right|\leq 1+c^Tx+b$