Wednesday, 14 October 2015

Lecture 3

The simplex algorithm is due to George Dantzig. In his paper on the origins of the simplex algorithm he writes:

"In the summer of 1947, when I first began to work on the simplex method for solving linear programs, the first idea that occurred to me is one that would occur to any trained mathematician, namely the idea of step by step descent (with respect to the objective function) along edges of the convex polyhedral set from one vertex to an adjacent one. I rejected this algorithm outright on intuitive grounds - it had to be inefficient because it proposed to solve the problem by wandering along some path of outside edges until the optimal vertex was reached. I therefore began to look for other methods which gave more promise of being efficient, such as those that went directly through the interior."

This is interesting because Dantzig says that the simplex method is something that any trained mathematician would think of - but that that on first sight it appears to be inefficient.

In practice these days computers solve problems with 100,000s constraints and variables. There is a nice on line simplex method tool which you might like to use to check answers that you obtain by hand when doing questions on Examples sheet 1. There is also a very nice program here, which runs under Windows.

It was written by Doug Kennedy at Trinity.  It takes the hard labour out of performing pivot operations and it may be configured to prompt the choice of pivot elements, or to solve problems automatically.