Mathematical Programming

, Volume 143, Issue 1–2, pp 87–110 | Cite as

Coordinated cutting plane generation via multi-objective separation

  • Edoardo Amaldi
  • Stefano Coniglio
  • Stefano Gualandi
Full Length Paper Series A

Abstract

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.

Keywords

Cutting plane methods Coordinated cutting plane generation Multi-objective separation 

Mathematics Subject Classification

90C10 Integer programming 90C29 Multi-objective and goal programming 90C57 Polyhedral combinatorics branch-and-bound branch-and-cut 

References

  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. C 1, 1–41 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Amaldi, E., Coniglio, S., Gualandi, S.: Improving cutting plane generation with 0–1 inequalities by bi-criteria separation. In: Festa, P. (ed.) Experimental Algorithms, Lecture Notes in Computer Science, pp. 266–275. Springer, Berlin (2010)Google Scholar
  3. 3.
    Andreello, G., Caprara, A., Fischetti, M.: Embedding \(\{0,1/2\}\)-cuts in a branch-and-cut framework: a computational study. INFORMS J. Comput. 19(2), 229–238 (2007)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58(1–3), 295–324 (1993)CrossRefMATHGoogle Scholar
  5. 5.
    Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0–1 programming by lift-and-project in a branch-and-cut framework. Manage. Sci. 42(9), 1229–1246 (1996)CrossRefMATHGoogle Scholar
  6. 6.
    Balas, E., Ceria, S., Cornuèjols, G., Natraj, N.: Gomory cuts revised. Oper. Res. Lett. 19, 1–9 (1996)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Balas, E., Fischetti, M., Zanette, A.: Lexicography and degeneracy: can a pure cutting plane algorithm work? Math. Program. A 1, 153–176 (2011)MathSciNetGoogle Scholar
  8. 8.
    Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. A 113, 219–240 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ben-Ameur, W., Neto, J.: Acceleration of cutting-plane and column generation algorithms: applications to network design. Networks 49, 3–17 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Chopra, S., Rao, M.R.: The Steiner tree problem I: formulations, compositions and extension of facets. Math. Program. 64, 209–229 (1994)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Chopra, S., Rao, M.R.: The Steiner tree problem II: properties and classes of facets. Math. Program. 64, 231–246 (1994)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Coniglio, S.: On coordinated cutting plane generation and mixed-integer programs with nonconvex 2-norm constraints. Ph.D. thesis, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy (2011)Google Scholar
  13. 13.
    Coniglio, S.: On coordinated cutting plane generation and mixed-integer programs with nonconvex 2-norm constraints. 4OR: Quart. J. Oper. Res. 1–2 (2012). doi:10.1007/s10288-012-0199-7
  14. 14.
    Cook, W., Fukusawa, R., Goycoolea, M.: Choosing the best cuts. In: 3rd Mixed Integer Programming, Workshop (2006)Google Scholar
  15. 15.
    Desrosiers, J., Lübbeke, M.: A primer in column generation. In: Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds.) Column Generation, GERAD 25th Anniversary, chap. 1, pp. 1–32. Springer, Berlin (2005)Google Scholar
  16. 16.
    Edmonds, J., Karp, R.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)CrossRefMATHGoogle Scholar
  17. 17.
    Fischetti, M., Lodi, A.: Optimizing over the first Chvàtal closure. Math. Program. A 110(1), 3–20 (2006)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Fischetti, M., Salvagnin, D.: An in-out approach to disjunctive optimization. In: Lodi, A., Toth, P. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Lecture Notes in Computer Science, pp. 136–140. Springer, Berlin (2010)Google Scholar
  19. 19.
    Gomory, R.: An algorithm for integer solutions to linear programs. In: Graves, R., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)Google Scholar
  20. 20.
    Gualandi, S., Malucelli, F.: Exact solution of graph coloring problems via constraint programming and column generation. INFORMS J. Comput. 24, 81–100 (2012)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Held, S., Cook, W., Sewell, E.: Safe lower bounds for graph coloring. In: Günlük, O., Woeginger, G. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, pp. 261–273. Springer, Berlin (2011)Google Scholar
  22. 22.
    Johnson, D., Trick, M.: Cliques, Coloring, and Satisfiability, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. American Mathematical Society, Providence (1996)Google Scholar
  23. 23.
    Koch, T., Martin, A.: Solving Steiner tree problems in graphs to optimality. Networks 32(3), 207–232 (1998)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Koch, T., Martin, A., Voß, S.: SteinLib: an updated library on steiner tree problems in graphs. In: Cheng, X., Du, D. (eds.) Steiner Trees in Industry, vol. 11, pp. 285–326. Kluwer, Dordrecht (2001)CrossRefGoogle Scholar
  25. 25.
    Nemhauser, G., Wolsey, L.: Integer Programming and Combinatorial Optimization. Wiley, New York (1980)Google Scholar
  26. 26.
    Ostergard, P.: A fast algorithm for the maximum clique problem. Disc. Appl. Math. 120(1–3), 197–207 (2002)Google Scholar
  27. 27.
    Phillips, N.: A weighting function for pre-emptive multicriteria assignment problems. J. Oper. Res. Soc. 38, 797–802 (1987)MATHGoogle Scholar
  28. 28.
    Wesselmann, F., Suhl, U.: Implementation techniques for cutting plane management and selection. Technical report, University of Paderborn, Paderborn (2007)Google Scholar
  29. 29.
    Williams, H.: Model Building in Mathematical Programming. Wiley, New York (1999)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  • Edoardo Amaldi
    • 1
  • Stefano Coniglio
    • 1
  • Stefano Gualandi
    • 1
    • 2
  1. 1.Dipartimento di Elettronica ed InformazionePolitecnico di MilanoMilanoItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly

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