The ellipsoid method has an undeniable historical relevance (due to its role in establishing polynomial time for linear programming with integer data). In addition, its underlying idea is simple and elegant. Unfortunately, it is not efficient in practice compared with both the simplex method and the more recent interior-point methods. In this chapter, we describe the latter in the context of linear programming.
Unlike the ellipsoid method, which seems tailored for feasibility problems, interior-point methods appear to be designed to solve optimization problems. In linear programming, however, it is possible to recast problems of one kind as problems of the other, and we take advantage of this feature to present an algorithmic solution for the feasibility problem PCFP. We see that again, the condition number Open image in new window of the data plays a role in the complexity of this solution.