How to find out Geometric Median

Question

The question is:

Given N points(in 2D) with x and y coordinates, find a point P (in N given points) such that the sum of distances from other(N-1)

Answer 1

You can solve the problem as a convex programming (The objective function is not always convex). The convex program can be solved using an iterative such as L-BFGS. The cost for each iteration is O(N) and usually the number of required iteration is not large. One important point to reduce the number of required iterations is that we know the optimum answer is one of the point in the input. So, the optimization can be stopped when its answer become close to one of the input points.