Friday, 16 October 2015

Lecture 4

On page 20 of the notes the first tableau should have $a_{21}=-2$. In fact it does in the notes, but not in the overhead slide I was using.

It is not obvious that Gomory's cutting plane algorithm should terminate. If interested, you can read this proof that it terminates.

The paper I mentioned about using linear programming to find a potential function needed for proofs in an online bin packing problem is

E.G. Coffman, Jr, D.S. Johnson, P.W. Shor and R. R. Weber. Markov chains, computer proofs, and average-case analysis of best fit bin packing. In Proc. 25 Annual ACM Symposium on Theory of Computing, San Diego, May, pages 412-421, 1993.

I had forgotten the date. Obviously computers are now much more powerful. It might be possible to extend the table in this paper, and perhaps formulate some conjectures concerning which $\{j,k\}$ lead to bounded or linear waste.

I mentioned that CPLEX is one of the leading commercial packages for linear programming. One can solve problems of medium size quite quickly with Mathematica. I have used Mathematica for problems of a few hundred variables and constraints.