On the convergence of the affine-scaling algorithm

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 is its convergence in the presence of degeneracy (which is important since 'practical" problems tend to be degenerate). 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-,/V, where n is the number of variables and P E (0, 1] is a certain stepsize ratio. For one particular stepsize choice which is an extension of that of Barnes, the limit point is shown to have a cost which is within O(,3) of the optimal cost and, for B sufficiently small, is shown to be 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, we show that it suffices to take 1 = I l where m is the number of constraints and C is the sum of the cost coefficients, to achieve exact optimality.