I have a convex optimization problem where the feasible region is defined as the convex hull of a set of vertices. Even though the vertex set is in the low dozens of points, finding the half-space (H-) representation of the polytope looks intractable. So I don't have a standard-format set of constraints to plug into the usual techniques.
Are there any transformations that would let me work around this and directly optimize over the V-representation?
I would be willing to do something like hill-climbing, but since I still have to test polytope-membership, I'm not sure if I gain anything.
Just in case it helps, the polytope is fairly degenerate (e.g. ~50-dimensional but in an ambient space of about 80 dimensions). It has a high degree of symmetry (although I don't know the symmetry group, I may have enough information to get it). My sketchy understanding is that the symmetry and degeneracy both make it harder to use standard techniques. It's also a 0/1-polytope (every vertex has either 0 or 1 for every coordinate.)
I could possibly use these facts to reduce the cost of getting the H-representation, but I don't know of any techniques for that and would be headed into unknown territory; I'd also appreciate literature pointers in that direction.