This question mentions how gradient descent might not be a good approach for this specific problem. It also mentions the only way to be sure is to mess around with it. I am wondering if this is true, is there no way to mathematically prove that there are a lot of local minima?
Surely there is a way to prove the function isn't convex and/or maybe one could graph certain contours to show it has many local minima.
In short, I want to be able to show that the following function
$$f(v) = \sum_{i,j} \left( \|v_i - v_j \|_2 - d_{i,j} \right)^2$$
where $v_i$ and $v_j$ are points, with their respective $x$ and $y$ variables, doesn't converge and presents a lot of local minima that might make the gradient descent very dependent on initial points. How would I go about it?