Journal of Optimization Theory and Applications

, Volume 91, Issue 3, pp 561–583 | Cite as

An exterior-point method for linear programming problems

  • J. F. Andrus
  • M. R. Schaferkotter
Contributed Papers

Abstract

This paper proves the convergence of an algorithm for solving linear programming problems inO(mn2) arithmetic operations. The method is called an exterior-point procedure, because it obtains a sequence of approximations falling outside the setU of feasible solutions. Each iteration consists of a single step within some constraining hyperplane, followed by one or more projections which force the new approximation to fall within some envelope aboutU. The paper also discusses several numerical applications. In some types of problems, the method is considerably faster than a standard simplex method program when the size of the problem is sufficiently large.

Key Words

Linear programming polynomial time algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrus, J. F.,An Exterior-Point Method for the Convex Programming Problem, Journal of Optimization Theory and Applications, Vol. 72, pp. 37–63, 1992.CrossRefGoogle Scholar
  2. 2.
    Gonzaga, C. C.,An Algorithm for Solving Linear Programming Problems in O(n 3L)Operations, Progress in Mathematical Programming: Interior and Related Methods, Edited by N. Megiddo, Springer Verlag, New York, New York, pp. 1–28, 1989.Google Scholar
  3. 3.
    Karmarkar, N.,A New Polynomial Time Algorithm for Linear Programming, Combinatorica, Vol. 4, pp. 373–395, 1984.Google Scholar
  4. 4.
    Khachian, L. G.,Polynomial Algorithms in Linear Programming, USSR Computational Mathematics and Mathematical Physics, Vol. 20, pp. 53–72, 1980.CrossRefGoogle Scholar
  5. 5.
    Monteiro, R. D. C., andAdler, I.,Interior Path Following Primal-Dual Algorithms, Part 1: Linear Programming, Mathematical Programming, Vol. 44, pp. 27–41, 1989.CrossRefGoogle Scholar
  6. 6.
    Renegar, J.,A Polynomial-Time Algorithm, Based on Newton's Method, for Linear Programming, Mathematical Programming, Vol. 40, pp. 59–93, 1988.CrossRefGoogle Scholar
  7. 7.
    Vaidya, P. M.,An Algorithm for Linear Programming Which Requires O(((m+n)n 2+(m+n)1.5 n)L)Arithmetic Operations, Mathematical Programming, Vol. 47, pp. 175–201, 1990.CrossRefGoogle Scholar
  8. 8.
    Domich, P. D., Hoffman, K. L., Jackson, R. H. F., Saunders, P. S., andShier, D. R.,Comparison of Mathematical Programming Software: A Case Study Using Discrete L 1-Approximation Codes, Computers and Operations Research, Vol. 14, pp. 435–447, 1987.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • J. F. Andrus
    • 1
  • M. R. Schaferkotter
    • 1
  1. 1.Department of MathematicsUniversity of New OrleansNew Orleans

Personalised recommendations