The Ellipsoid Algorithm

  • Howard Karloff
Part of the Modern Birkhäuser Classics book series (MBC)


The first polynomial-time linear programming algorithm, the Ellipsoid Algorithm was constructed by Soviet mathematicians, L. G. Khachiyan providing the final details in 1979. It is sometimes known as Khachiyan’s Algorithm to acknowledge Khachiyan’s contribution. The algorithm differs radically from the Simplex Algorithm, in that it almost completely ignores the combinatorial structure of linear programming


Convex Hull Linear Inequality Orthogonal Matrix Affine Transformation Basic Feasible Solution 
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  1. [7]
    B. Aspvall and R. E. Stone, “Khachiyan’s Linear Programming Algorithm,” Journal of Algorithms 1 (1980), 1-13.MATHCrossRefMathSciNetGoogle Scholar
  2. [12]
    R. G. Bland, D. Goldfarb and M. J. Todd, “The Ellipsoid Method: A Survey,” Operations Research 29 (1981), 1039-1091.MATHCrossRefMathSciNetGoogle Scholar
  3. [19]
    P. Gacs and L. Lovasz, “Khachiyan’s Algorithm for Linear Programming,” Mathematical Programming Study 14(1981), 61-68.Google Scholar
  4. [26]
    C. C. Gonzaga, “An Algorithm for Solving Linear Programming Problems in O(n 3 L)Operations,” in Progress in Mathematical Programming: Interior-Point and Related Methods, N. Megiddo, ed., Springer-Verlag, New York, NY, 1989, 1-28.Google Scholar
  5. [27]
    M. Grotschel, L. Lovasz and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York, NY, 1988.Google Scholar
  6. [33]
    D. B. Judin and A. S. Nemirovskii, “Informational Complexity and Effective Methods for the Solution of Convex Extremal Problems” (in Russian), Ekonomika; Matematich-eskie Metody 12 (1976), 357-369.MathSciNetGoogle Scholar
  7. [34]
    N. Karmarkar, “A New Polynomial-Time Algorithm for Linear Programming,” Combinatorica 4 (1984), 373-395.MATHCrossRefMathSciNetGoogle Scholar
  8. [35]
    L. G. Khachiyan, “A Polynomial Algorithm for Linear Programming” (in Russian), Doklady Akademiia Nauk USSR 244 (1979), 1093-1096. A translation appears in Soviet Mathematics Doklady 20 (1979), 191-194.MATHMathSciNetGoogle Scholar
  9. [40]
    E. L. Lawler, “The Great Mathematical Sputnik of 1979,” The Mathematical Intelligencer 2(1980), 191-198.MATHCrossRefGoogle Scholar
  10. [42]
    R. D. C. Monteiro and I. Adler, “Interior Path Following Primal-Dual Algorithms. Part I: Linear Programming,” Mathematical Programming 44 (1989), 27-41.MATHCrossRefMathSciNetGoogle Scholar
  11. [44]
    C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice Hall, Engle-wood Cliffs, NJ, 1982.MATHGoogle Scholar
  12. [45]
    J. Renegar, “A Polynomial-Time Algorithm, Based on Newton’s Method, for Linear Programming,” Mathematical Programming 40 (1988), 50-94.Google Scholar
  13. [46]
    W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, NY, 1987, 54-55.Google Scholar
  14. [47]
    A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, NY, 1986.MATHGoogle Scholar
  15. [50]
    N. Z. Shor, “Utilization of the Operation of Space Dilation in the Minimization of Convex Functions” (in Russian), Kiber-netika 1 (1970), 6-12. A translation appears in Cybernetics 6 (1970), 7-15.Google Scholar
  16. [57]
    E. Tardos, “A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs,” Operations Research 34 (1986), 250-256.MATHCrossRefMathSciNetGoogle Scholar
  17. [60]
    P. M. Vaidya, “An Algorithm for Linear Programming Which Requires O(((m+ n)n 2 +(m + n)1.5 n)L) Arithmetic Operations,” Mathematical Programming 47 (1990), 175-201.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Howard Karloff
    • 1
  1. 1.College of Computing, Georgia TechAtlantaUSA

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