Reformulating a Euclidean distance minimization problem into a semidefinite program The following minimization problem is a Euclidean distance form of a single-facility location problem
$$\min \quad \sum_j \sqrt {(x-a_j)^2+(y-b_j)^2}$$
where $(x,y)$ and $(a_j,b_j)$ are the coordinates of the new facility and  current facilities, respectively.
I mistakenly tried to reformulate it as a second-order conic program (SOCP) and found that it is not possible. I wonder, is it possible to reformulate it as a convex program using semidefinite cones?

It's SOCP representable, in fact it almost doesn't get more SOCP than this.
For instance, minimizing  $||q|| + ||p||$ is equivalent to minimizing $s + t$ subject to $||q||\leq t, ||p||\leq s$ which is an SOCP in standard form.
