CSC 2411 - Linear Programming and Combinatorial Optimization ∗ Lecture 6 : The Ellipsoid Method

This lecture introduces the Ellipsoid Method, the first polynomialtime algorithm to solve LP. We start by discussing the historical significance of its discovery by L. Khachiyan. Next, we argue that the ability to decide the feasibility of a version of the constraints of an LP is as hard as solving the LP. To gather the intuition behind the Ellipsoid Method, we draw an analogy with a problem of finding a large creature in a constrained space. Through this analogy, we formulate 3 “ingredients” that we require of the potential algorithm that will find this creature. Next, we show that each of these ingredients exists for the feasibility problem we are solving. Finally, we discuss the need for ellipsoids as the bounding volumes used to locate the feasible set. As a result, we are able to sketch the pseudocode of the Ellipsoid Method, whose correctness and polynomial time complexity is demonstrated. As preparation for the upcoming lecture, we consider expressing an ellipsoid using a positive semi-definite matrix.