I am trying to minimize a convex objective numerically using gradient descent. I select the starting point randomly. I repeat the experiment multiple times. The optimal objective value I get each time is quit the same, but the minimizer is very different. Is it natural? How should it be handled in experiments?
Yes, this is entirely possible. The fact that a problem is convex just guarantees that any local minimum is a global minimum. You can have many points that reach this global minimum. Intuitively, you can think of a function with a "flat" bottom, somewhat like a bath tub, so two points can be far away but still be minimal.