I am looking for an algorithm to find the point that minimizes the sum of the great circle distances to a set of fixed points on a sphere. In more detail: Given $x_1,\ldots, x_k$ fixed points in the $\mathbb{R}^n$ sphere. All of them have non-negative coefficients, i.e., $\|x_i\| = 1$ and $x_{ij} \geq 0$. I wish an algorithm to find the point $y$ such that $\|y\|=1$ minimizing $\sum_{i=1}^k d_{x_i}(y)$, where $d_{x_i}(y) = \arccos( \langle x,y \rangle)$.I know this is considered a facility location problem but I can't find any algorithm that solves this particular problem.
Facility location on a Sphere with great circle distance
algorithms;optimization;linear programming