TY - JOUR

T1 - On the convergence of the affine-scaling algorithm

AU - Tseng, Paul

AU - Luo, Zhi Quan

N1 - Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1992/10/5

Y1 - 1992/10/5

N2 - 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 {Mathematical expression}, where n is the number of variables and β ∈ (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, for β 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 take β = 1/(6 mC), where m is the number of constraints and C is the sum of the cost coefficients, to attain exact optimality.

AB - 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 {Mathematical expression}, where n is the number of variables and β ∈ (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, for β 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 take β = 1/(6 mC), where m is the number of constraints and C is the sum of the cost coefficients, to attain exact optimality.

KW - affine-scaling

KW - ergodic convergence

KW - Linear program

UR - http://www.scopus.com/inward/record.url?scp=0027112829&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027112829&partnerID=8YFLogxK

U2 - 10.1007/BF01580904

DO - 10.1007/BF01580904

M3 - Article

AN - SCOPUS:34249836074

VL - 56

SP - 301

EP - 319

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 3

ER -