The worst-case step in Karmarkar’s algorithm

In this note the worst-case performance in a single step of Karmarkar’s projective algorithm for linear programming is considered. In the transformed problem which arises on each iteration it is shown that the critical ratio ‘r/R’ can be improved (asymptotically) by a factor of two. It is also shown that in the original problem, where performance is characterized by reduction in the potential function, the worst-case reduction can be improved to approximately 0.72. Moreover, it is demonstrated that both of these bounds are tight, so that no further improvement based on the analysis of a single step is possible.