Reformulating a Euclidean distance minimization problem into a semidefinite program

Question

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?

Answer 1

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.