I would like to find an optimal (e.g. like in a least square manner) solution for the following system of equations (noisy distance measurements between 4 points):
$ (\color{green}{x_1}-x_2)^2+(\color{green}{y_1-y_2})^2=(\color{green}{d_{1,2}})^2$
$ (\color{green}{x_1}-x_3)^2+(\color{green}{y_1}-y_3)^2=(\color{green}{d_{1,3}})^2$
$ (\color{green}{x_1}-x_4)^2+(\color{green}{y_1}-y_4)^2=(\color{green}{d_{1,4}})^2$
$ (x_2-x_3)^2+(\color{green}{y_2}-y_3)^2=(\color{green}{d_{2,3}})^2$
$ (x_2-x_4)^2+(\color{green}{y_2}-y_4)^2=(\color{green}{d_{2,4}})^2$
$ (x_3-x_4)^2+(y_3-y_3)^2=(\color{green}{d_{3,4}})^2$
Let's assume that the green are the known ones (with $x_1, y_1, y_2 = 0$) and the black are the unknown ones. If there is no single solution (due to noise in the measurements), how should I approach the problem? I also reckon that setting $x_1,y_1,y_2 = 0$ is not enough since the system solution could still be a reflection about the segment formed by these two points, but I do not mind accepting any of the two solutions.
Lastly, let me mention that my system involves in fact $20$ points (i.e. $190$ equations with $37$ unknowns)