Monday, 26 October 2015

Lecture 7

You will have seen that the details of the Ellipsoid algorithm are really quite intricate. I have tried to show you the key ideas in Lemmas 7.1-7.3, but have not dealt with the issue of approximating square roots in polynomial running time. If you have a question about the explanation or think you see an error in the notes, please let me know so I can fix it.

I think the proof of Lemma 7.3 is particularly cute for the way it uses the primal-dual theory of linear programming to show that $\text{P}=\emptyset\implies \text{P}_\epsilon=\emptyset$. Question 10 on Examples Sheet 1 is similar in spirit.

Remember that the importance of the Ellipsoid algorithm lies in its theoretical rather than practical. In practice, linear programs are solved by high quality implementations of the simplex algorithm or by an interior point method like Karamakar's method.

In Lemma 7.2 we used the fact that $Q\subset \mathbb{R}^n$, the convex hull of $n+1$ vectors, $v^0,v^1,\dotsc,v^n$, that do not lie in the same hyperplane has
$$ \text{vol}(Q)=\frac{1}{n!}\left| \det \left( \begin{array}{ccc} 1 & \cdots & 1 \\ v^0 & \cdots & v^n \end{array} \right)\right|. $$ The proof of this result is not for our course, but if you are interested see A Note on the Volume of a Simplex, P. Stein The American Mathematical Monthly Vol. 73, No. 3 (Mar., 1966), pp. 299-301