Mathematical Programming

, Volume 40, Issue 1–3, pp 59–93 | Cite as

A polynomial-time algorithm, based on Newton's method, for linear programming

  • James Renegar


A new interior method for linear programming is presented and a polynomial time bound for it is proven. The proof is substantially different from those given for the ellipsoid algorithm and for Karmarkar's algorithm. Also, the algorithm is conceptually simpler than either of those algorithms.

Key words

Linear programming interior method computational complexity Newton's method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Aho, J. Hopcroft and J. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1974).Google Scholar
  2. [2]
    D.A. Bayer and J.C. Lagarias, “The non-linear geometry of linear programming, I. Affine and projective scaling trajectories, II. Legendre transform coordinates, III. Central trajectories,” preprints, AT&T Bell Laboratories (Murray Hill, NJ, 1986).Google Scholar
  3. [3]
    L. Blum, talk at Workshop on Problems Relating Numerical Analysis to Computer Science, Mathematical Sciences Research Institute, Berkeley, California (January 1986).Google Scholar
  4. [4]
    L. Blum, “Towards an asymptotic analysis of Karmarkar's algorithm,”Information Processing Letters 23 (1986) 189–194.Google Scholar
  5. [5]
    P. Huard, “Resolution of mathematical programming with non-linear constraints by the method of centers,” in: J. Abadie, ed.,Non-Linear Programming (North-Holland, Amsterdam, 1967) pp. 207–219.Google Scholar
  6. [6]
    N. Karmarkar, “A new polynomial-time algorithm for linear programming,” in:Proceedings of the 16th Annual ACM Symposium on Theory of Computing (1984), ACM, New York, 1984, pp. 302–311; revised version:Combinatorica 4 (1984) pp. 373–395.Google Scholar
  7. [7]
    L.G. Khachiyan, “A polynomial algorithm in linear programming,”Soviet Mathematics Doklady 20 (1979) pp. 191–194.Google Scholar
  8. [8]
    L.G. Khachiyan, “Polynomial algorithms in linear programming,”USSR Computational Mathematics and Mathematical Physics 20 (1980) pp. 53–72.Google Scholar
  9. [9]
    J. Lagarias, talk at Mathematical Sciences Research Institute (Berkeley, California, December, 1985).Google Scholar
  10. [10]
    N. Megiddo and M. Shub, “Boundary behavior of interior point algorithms in linear programming,” Research Report RJ5319, IBM Thomas J. Watson Research Center (1986).Google Scholar
  11. [11]
    S. Smale, “On the efficiency of algorithms of analysis,”Bulletin of the American Mathematical Society 13 (1985) pp. 87–121.Google Scholar
  12. [12]
    S. Smale, “Algorithms for solving equations,” to appear in:Proceedings, International Congress of Mathematicians (Berkeley, 1986).Google Scholar
  13. [13]
    Gy. Sonnevend, “A new method for solving a set of linear (convex) inequalities and its applications for identification and optimization,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University, 1088, Budapest, Muzeum Körut 6–8.Google Scholar
  14. [14]
    Gy. Sonnevend, “An analytical centre' for polyhedrons and new classes of global algorithms for linear (smooth convex) programming,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University, 1088, Budapest, Muzeum Körut 6–8.Google Scholar
  15. [15]
    P. Vaidya, “An algorithm for linear programming which requires O((m+n)n 2+(m+n)1.5 n)L) arithmetic operations,” AT&T Bell Laboratories, Murray Hill, NJ (1987).Google Scholar
  16. [16]
    J.H. Wilkinson,The algebraic Eigenvalue Problem (Oxford University Press, Oxford, 1965).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1988

Authors and Affiliations

  • James Renegar
    • 1
  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

Personalised recommendations