The ellipsoid method repeatedly asks a separation oracle about the center of a containing ellipsoid and replaces that ellipsoid by a smaller one containing its intersection with a separating half-space. The volume decreases by a dimension-dependent factor at each iteration. For rational polyhedra, the exact feasibility theorem has polynomial bit complexity in the dimension and coefficient encoding bounds when separation is polynomial time. It includes lower-dimensional feasible sets through rational precision bounds or suitable relaxations; one must not declare infeasibility just because the original feasible set has zero volume.
New to topics? Read the docs here!