So my knowledge in linear programming is quite limited. But I do have an intuition why the simplex method (which in my mind is a greedy GD method) would fail for large number of extreme points.
I am facing a linear optimization problem in which the convex set of possible solutions is not a polytope but rather a round object. My first attempt was to approximate this by a polytope with a large number of extreme points and then run the simplex method. However this takes $long$ for $good$ approximations.
Does anyone know a better method to solve the problem? I would also appreciate a good, thorough source be it a textbook, paper, lecture note, video lecture, etc.