The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into two parts, and then by querying (some sort of) center of the current feasible set, the volume of the new feasible set decreases geometrically. The decay depends on the choice of center.
Question 1: In most of the methods for choosing center, we do not have any control on the diameter of the new feasible set. We can only argue that the volume decreases. Is there any method for choosing center that guarantees a decrease in the diameter of the new feasible set?
Question 2: Is there a class of convex functions such that running cutting plane method on them ensures a decrease in the diameter of the new feasible set?
Question 3: Is there a prior work on using stochastic gradient instead of exact gradient for cutting plane methods?
