On the convergence of the affine-scaling algorithm

Abstract

The affine-scaling algorithm, first proposed by Dikin, is presently enjoying great popularity as a potentially effective means of solving linear programs. An outstanding question about this algorithm concerns its convergence in the presence of degeneracy. In this paper, we give new convergence results for this algorithm that do not require any non-degeneracy assumption on the problem. In particular, we show that if the stepsize choice of either Dikin or Barnes or Vanderbei, et al. is used, then the algorithm generates iterates that converge at least linearly with a convergence ratio of\(1 - \beta /\sqrt n \), wheren is the number of variables andβ ∈ (0,1] is a certain stepsize ratio. For one particular stepsize choice which is an extension of that of Barnes, we show that the cost of the limit point is within O(β/(1−β)) of the optimal cost and, forβ sufficiently small (roughly, proportional to how close the cost of the nonoptimal vertices are to the optimal cost), is exactly optimal. We prove the latter result by using an unusual proof technique, that of analyzing the ergodic convergence of the corresponding dual vectors. For the special case of network flow problems with integer data, we show that it suffices to takeβ = 1/(6mC), wherem is the number of constraints andC is the sum of the cost coefficients, to attain exact optimality.