On the improvement per iteration in Karmarkar's algorithm for linear programming

Abstract

We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameterα (so that we must have 0 <α ≤ 1) the best possible guaranteeδ _n (α) inn dimensional space is essentially ln 2 ≃ 0.69; and to achieve this we must takeα about 1. Indeed we show the precise result thatδ _n (α) equals ln(1 +α)-ln(1 −α/(n − 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to aboutδ ^* ≃ 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.