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Integral Simplex Methods for the Set Partitioning Problem: Globalisation and Anti-Cycling

  • Elina Rönnberg
  • Torbjörn Larsson
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 141)

Abstract

The set partitioning problem is a generic optimisation model with many applications, especially within scheduling and routing. It is common in the context of column generation, and its importance has grown due to the strong developments in this field. The set partitioning problem has the quasi-integrality property, which means that every edge of the convex hull of the integer feasible solutions is also an edge of the polytope of the linear programming relaxation. This property enables, in principle, the use of solution methods that find improved integer solutions through simplex pivots that preserve integrality; pivoting rules with this effect can be designed in a few different ways. Although seemingly promising, the application of these approaches involves inherent challenges. Firstly, they can get be trapped at local optima, with respect to the pivoting options available, so that global optimality can be guaranteed only by resorting to an enumeration principle. Secondly, set partitioning problems are typically massively degenerate and a big hurdle to overcome is therefore to establish anti-cycling rules for the pivoting options available. The purpose of this chapter is to lay a foundation for research on these topics.

Keywords

Quasi-integrality Set partitioning Integral simplex method Anti-cycling rules 

References

  1. 1.
    Balas, E., Padberg, M.W.: On the set-covering problem. Oper. Res. 20, 1152–1161 (1972)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hellstrand, J., Larsson, T., Migdalas, A.: A characterization of the uncapacitated network design polytope. Oper. Res. Lett. 12, 159–163 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hoffman, K., Padberg, M.: Set covering, packing and partitioning problems. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2008)Google Scholar
  4. 4.
    Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53, 1007–1023 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Murty, K.G.: Linear Programming. Wiley, New York (1983)zbMATHGoogle Scholar
  6. 6.
    Rönnberg, E., Larsson, T.: Column generation in the integral simplex method. Eur. J. Oper. Res. 192, 333–342 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rönnberg, E., Larsson, T.: All-integer column generation for set partitioning: basic principles and extensions. Eur. J. Oper. Res. 233, 529–538 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sastry, T.: One and two facility network design revisited. Ann. Oper. Res. 108, 19–31 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Thompson, G.L.: An integral simplex algorithm for solving combinatorial optimization problems. Comput. Optim. Appl. 22, 351–367 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Trubin, V.A.: On a method of solution of integer linear programming problems of a special kind. Translated by V. Hall. Soviet Math. Doklady 10, 1544–1546 (1969)zbMATHGoogle Scholar
  11. 11.
    Wilhelm, W.E.: A technical review of column generation in integer programming. Optim. Eng. 2, 159–200 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wolsey, L.A.: Integer Programming. Wiley, New York (1998)zbMATHGoogle Scholar
  13. 13.
    Yemelichev, V.A., Kovalev, M.M., Kravtsov, M.K.: Polytopes, Graphs and Optimisation (Translated by G. H. Lawden). Cambridge University Press, Cambridge (1984)Google Scholar
  14. 14.
    Zaghrouti, A., Soumis, F., El Hallaoui, I.: Integral simplex using decomposition for the set partitioning problem. Oper. Res. 62, 435–449 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zörnig, P.: Degeneracy Graphs and Simplex Cycling. Springer, Berlin (1991)CrossRefGoogle Scholar
  16. 16.
    Zörnig, P.: Systematic construction of examples for cycling in the simplex method. Comput. Oper. Res. 33, 2247–2262 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden

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