The Worst-Case Step in Karmarkar's Algorithm

In this note we consider the worst-case performance in a single step of Karmarkar's projective algorithm for linear programming. In the transformed problem which arises on each iteration we show that the critical ratio “r/R” can be improved (asymptotically) by a factor of two. We also show 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, we demonstrate that both of these bounds are tight, so that no further improvement based on the analysis of a single step is possible.