SOCP formulation from a convex constraint

Question

Suppose I have the constraint set

$$\{x: a_1\|x-b_1\|_2-a_2\|x-b_2\|_2\leq t\}$$

where $a_1,a_2\geq 0$. I know that the function

$$f(x)=a_1\|x-b_1\|_2-a_2\|x-b_2\|_2$$

is convex (it is given that $f$ is convex) and, thus, any single constraint in the above set is an epigraph and, thus, convex. How can I reformulate the above constraint into an SOCP?

Is it correct to write it as follows?

$$\{x,w,v; a_1w-v\leq t, \|x-b_1\|_2\leq w, \|x-b_2\|_2\leq v\}$$

I am unsure about this as there is a negative sign with $v$. Any help would be really great.

Answer 1

Is it correct to write it as:

$ \left\{x,w,v \; | \; a_{1}w−a_{2}v \leq t, \|x−b_{1} \| \leq w, \| x-b_{2} \| \leq v \right\}$.

No, this doesn't work. For example, if $a_{1}=4$, $b_{1}=4$, $a_{2}=0.2$, $b_{2}=0$, then

$x=1000$, $t=0$ doesn't satisy the original constraint, but

$x=1000$, $t=0$, $w=1000$, $v=1000000$, satisfies the proposed equivalent constraints.