Advertisement

The Homogeneous Self-Dual Method

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 196)

Abstract

In Chapter 18, we described and analyzed an interior-point method called the path-following algorithm. This algorithm is essentially what one implements in practice but as we saw in the section on convergence analysis, it is not easy (and perhaps not possible) to give a complete proof that the method converges to an optimal solution. If convergence were completely established, the question would still remain as to how fast is the convergence. In this chapter, we shall present a similar algorithm for which a complete convergence analysis can be given.

Bibliography

  1. Adler, I., Karmarkar, N., Resende, M., and Veiga, G. (1989). An implementation of Karmarkar’s algorithm for linear programming. Mathematical Programming, 44, 297–335.CrossRefGoogle Scholar
  2. Mehrotra, S. (1989). Higher order methods and their performance (Techincal Report TR 90-16R1) Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston. Revised July 1991.Google Scholar
  3. Mehrotra, S. (1992). On the implementation of a (primal-dual) interior point method. SIAM Journal on Optimization, 2, 575–601.CrossRefGoogle Scholar
  4. Mizuno, S., Todd, M., and Ye, Y. (1993). On adaptive-step primal-dual interior-point algorithms for linear programming. Mathematics of Operations Research, 18, 964–981.CrossRefGoogle Scholar
  5. Tucker, A. (1956). Dual systems of homogeneous linear equations. Annals of Mathematics Studies, 38, 3–18.Google Scholar
  6. Xu, X., Hung, P., and Ye, Y. (1993). A simplified homogeneous and self-dual linear programming algorithm and its implementation (Techincal Report). College of Business Administration, University of Iowa. To appear in Annals of Operations Research.Google Scholar
  7. Ye, Y., Todd, M., and Mizuno, S. (1994). An \(o(\sqrt{n}l)\)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19, 53–67.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations