I have $N$ unknown 2D points ${\bf x} = (x,y)$ and I want to minimize:
$f({\bf x}) = \sum_{i=0}^{N-1}{(||x_{i+1} - x_i||_2 - c_i)^2}$
where c are $N-1$ known scalars. What is the simplest form that I can use to solve this? I was thinking to try to express it as a quadratic programming problem by solving for the homogeneous coordinates of the points, and in the end divide by the $3^{rd}$ coordinate:
${\bf x}^TQ{\bf x} + {\bf c}^T{\bf x}$
where ${\bf x}$ is now a $3N\times 1$ vector. The reason the minimization expands to quadratic polynomials with two variables, in the form $ax^2+bxy+cy^2+dx+ey+f$, where $b=0$.
Would this work? If yes, is there any tricky bit that I should be aware of?