How to reformulate ${\left\| {Ax + b} \right\|^2} \le c + {b^T}b - r$ as an SOCP constraint?

Question

I am running into a constrain that looks quite similar but not really in the standard form of an SOCP. That is:

$${\left\| {Ax + b} \right\|^2} \le c + {b^T}b - r$$

Standard form of SOCP:

$$\left\| {Ax + b} \right\| \le {c^T}x + d$$

Is it possible to reformulate this constraint as SOCP ? And if so how can I do it ?

Answer 1

This is the typical reformulation when the right hand side of your constraint is not a parameter: $$\left\lVert \begin{pmatrix}2(Ax+b) \\ c+b^Tb-r - 1 \end{pmatrix} \right\rVert \leq c+b^Tb-r + 1$$