Wednesday, 18 November 2015

Lecture 17

It was a pity my coloured figure to explain Sperner's lemma in $\mathbb{R}^2$ did not show up well on the overhead projector. On page 82 of the notes I have now put a version with  colours black, white and grey. I hope you can understand the lemma and its proof when you review this. You should see that the proof of the Brouwer fixed point theorem (via Sperner's lemma) and the Lemke-Howson algorithm use a common idea that a graph in which all vertices have degree 1 or 2 must have an even number of vertices of degree 1, and that given one vertex of degree 1 it is possible to find another by following a unique path.

Along each side of the large triangle there is an odd number of edges whose two endpoints are differently coloured. Entering along the base across such an edge and following a path crossing edges which are similarly coloured, we must, with at least one choice of initial entry edge, reach a rainbow triangle. Notice that this also proves that the number of rainbow triangles is odd. In this example there are 3 rainbow triangles. Only one is found by a following a path entering at the base. What happens in the above if you try to enter along an edge on the long right side whose endpoints are red and blue?

Notice that a linear programming problem can be formulated as linear complementarity problem and solved using Lemke's algorithm. (Quadratic programming with $D=0$.)

I have made some corrections and improvements at the bottom of page 80 of the notes. Here $w,z$ needed to be swapped, and I have tried to improve the explanation.

The formulation of a knapsack problem as a linear complementarity problem in 17.3 I took from the paper On the solution of NP-hard linear complementarity problems by Judice, Faustino and Martins Ribeiro, 2002. The authors conclude that Lemke's algorithm is not a good way to solve this LCP.