Computational Optimization and Applications

, Volume 2, Issue 4, pp 299–316 | Cite as

The ellipsoid algorithm using parallel cuts

  • Aiping Liao
  • Michael J. Todd


We present an ellipsoid algorithm using parallel cuts which is robust and conceptually simple. If the ratio fo the distance between the parallel cuts under consideration and the corresponding radius of the current ellipsoid is less than or equal to some constant, it is called the “canonical case.” Applying our algorithm to this case the volume of the next ellipsoid decreases by a factor which is, at worst, exp\(\left( { - \frac{1}{{2(n + 2)}}} \right).\) For the noncanonical case, we first add an extra constraint to make it a canonical case in a higher-dimensional space, then apply our algorithm to this canonical case, and finally reduce it back to the original space. Some interesting variants are also presented to show the flexibility of our basic algorithm.


Ellipsoid algorithm parallel cuts complexity linear system 


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  1. 1.
    M. Avriel,Nonlinear Programming: Analysis and Methods, Prentice-Hall, Inc: Englewood Cliffs, NJ, 1976.Google Scholar
  2. 2.
    B.P. Burrell and M.J. Todd, “The ellipsoid method generates dual variables,”Math. Oper. Res. vol. 10, pp. 688–700, 1985.Google Scholar
  3. 3.
    P. Gács and L. Lovász, “Khachiyan's algorithm for linear programming,”Math. Prog. Study, vol. 14, pp. 61–68, 1981.Google Scholar
  4. 4.
    M. Grötschel, L. Lovász, and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica, vol. 1, pp. 169–197, 1981.Google Scholar
  5. 5.
    R.A. Horn and C.R. Johnson,Matrix Analysis, Cambridge University Press: Cambridge, England, 1990.Google Scholar
  6. 6.
    A.S. Householder,The Theory of Matrices in Numerical Analysis, Ginn (Blaisdell): Boston, MA, 1964.Google Scholar
  7. 7.
    L.G. Khachian, “Polynomial algorithms for linear programming,”USSR Comput. Math. Math. Phys. vol. 20, pp. 53–72, 1980.Google Scholar
  8. 8.
    A. Liao, “Algorithms for Linear Programming via Weighted Centers,” PhD thesis, Cornell University, Ithaca, NY, 1992.Google Scholar
  9. 9.
    A. Liao and M.J. Todd, “Solving LP Problems via Weighted Centers,” Technical Report CTC93TR-145, Advanced Computing Research Institute, Cornell University, Ithaca, NY, 1993.Google Scholar
  10. 10.
    Y.E. Nesterov and A.S. Nemirovsky,Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms, SIAM Publications: Philadelphia, PA, 1993.Google Scholar
  11. 11.
    M.J. Todd, “On minimum volume ellipsoids containing part of a given ellipsoid,”Math. Oper. Res., vol. 7, pp. 253–261, 1980.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Aiping Liao
    • 1
  • Michael J. Todd
    • 1
  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthaca

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