Solving Linear Ordering Problems with a Combined Interior Point/Simplex Cutting Plane Algorithm

  • John E. Mitchell
  • Brian Borchers
Part of the Applied Optimization book series (APOP, volume 33)


We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primal-dual interior point method to solve the first few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm with one that uses only an interior point method and with one that uses only a simplex method. We solve integer programming problems with as many as 31125 binary variables. Computational results show that the combined approach can dramatically outperform the other two methods.


Interior Point Simplex Method Interior Point Method Cutting Plane Current Relaxation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Applegate, D., and R. Bixby and V. Chvátal and W. Cook. “The traveling salesman problem”, DIMACS, Rutgers University. New Brunswick, NJ, 1994.Google Scholar
  2. [2]
    Bahn, O., and O. Du Merle and J.L. Goffin and J.R Vial”. “A cutting plane method from analytic centers for stochastic programming”, Mathematical Programming. 69, 45–73, 1995.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Christof, T., and G. Reinelt. “Low-dimensional linear ordering polytopes”, IWR Heidelberg, Germany, 1997. Google Scholar
  4. [4]
    Christof, T., and G. ReineJt. “Algorithmic aspects of using small instance relaxations in parallel branch-and-cut”, IWR Heidelberg, Germany, 1998. Google Scholar
  5. [5]
    Czyzyk, J., and S. Mehrotra and M. Wagner and S.J. Wright. “PCx user guide (version 1.1)”, Optimization Technology Center, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439, November 1997.
  6. [6]
    Fishburn, P.C. “Induced binary probabilities and the linear ordering polytope: A status report”, Mathematical Social Sciences, 23, 67–80, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Goffin, J.-L., and J. Gondzio and R. Sarkissian and J.-P. Vial. “Solving nonlinear multicommodity network flow problems by the analytic center cutting plane method”, Mathematical Programming, 76, 131–154, 1997.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Gondzio, J. “HOPDM (ver 2.12) — A fast LP solver based on a primal-dual interior point method”, European Journal of Operational Research, 85, 221–225, 1995.zbMATHCrossRefGoogle Scholar
  9. [9]
    Gondzio, J. “Warm Start of the Primal-Dual Method Applied in the Cutting Plane Scheme”, Mathematical Programming, 83. 125–143, 1998. Scholar
  10. [10]
    Gondzio, J., and R. Sarkissian. “Column generation with a primal-dual method”, Logilab, HEC Geneva, Section of Management Sciences, University of Geneva, 102 Bd Carl Vogt, CH-1211 Geneva 4, Switzerland, June 1996. Scholar
  11. [11]
    Grötschel, M., and M. Jünger and G. Reinelt. “A cutting plane algorithm for the linear ordering problem”, Operations Research, 32, 1195–1220, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Grötschel, M., and M. Jünger and G. Reinelt. “Optimal triangulation of large real-world input-output matrices”, Statistiche Hefte, 25, 261–295, 1984.CrossRefGoogle Scholar
  13. [13]
    Jünger, M. “Polyhedral Combinatorics and the Acyclic Subdigraph Problem”, Heldermann, Berlin, 1985.zbMATHGoogle Scholar
  14. [14]
    Karp, R.M. “Reducibility among combinatorial problems”, R. E. Miller and J. W. Thatcher (eds.) Complexity of Computer Computations, Plenum Press, New York, 85–103, 1972.CrossRefGoogle Scholar
  15. [15]
    Leung, J., and J. Lee. “More Facets From Fences For Linear Ordering And Acyclic Subgraph Polytopes”, Discrete Applied Mathematics, 50, 185–200, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Mitchell, J.E. “Computational experience with an interior point cutting plane algorithm”, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180–3590, February 1997. Revised: April 1997. Available at: Scholar
  17. [17]
    Mitchell, J.E. “An interior point cutting plane algorithm for Ising spin glass problems”, Operations Research Proceedings, SOR 1997, Jena, Germany, P. Kischka and H.-W. Lorenz (eds). Springer-Verlag, 114–119, 1998. Scholar
  18. [18]
    Mitchell, J.E., and B. Borchers. “A primal-dual interior point cutting plane method for the linear ordering problem”, COAL Bulletin, 21, 13–18, November 1992.Google Scholar
  19. [19]
    Mitchell, J.E., and B. Borchers. “Solving real-world linear ordering problems using a primal-dual interior point cutting plane method”, Annals of Operations Research, 62, 253–276, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Mitchell, J.E., and M.J. Todd. “Solving combinatorial optimization problems using Karmarkar’s algorithm”, Mathematical Programming, 56, 245–284, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Reinelt, G. “The Linear Ordering Problem: Algorithms and Applications”, Heldermann, Berlin, 1995.Google Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • John E. Mitchell
  • Brian Borchers

There are no affiliations available

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