Skip to main content
SpringerLink
Log in
Book cover

Optimization in the Real World pp 37–53Cite as

Structure-Based Primal Heuristics for Mixed Integer Programming

  • Conference paper
  • First Online:

  • 1767 Accesses

Part of the book series: Mathematics for Industry ((MFI,volume 13))

Abstract

Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They help to reach optimality faster and provide good feasible solutions early in the solving process. In this paper, we present two new primal heuristics which take into account global structures available within MIP solvers to construct feasible solutions at the beginning of the solving process. These heuristics follow a large neighborhood search (LNS) approach and use global structures to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. The definition of the neighborhood is done by iteratively fixing variables and propagating these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. The neighborhood is solved as a sub-MIP and solutions are transferred back to the original problem. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about every third instance and therewith help to improve the average solving time.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    For a definition and discussion of the shifted geometric mean, see [1, Appendix A]. We use shifts of 10 and 100 for time and nodes, respectively.

  2. 2.

    We compute the average primal gap by means of the primal integral [7] as \(\frac{P(\text {t}_{\text {max}})}{\text {t}_{\text {max}}}\) with \(\text {t}_{\text {max}}= 3600\) s, \(P(x) = \int _{t = 0}^{x} \gamma (t)\,dt\) and \(\gamma (t)\) the primal gap at time t.

References

  1. Achterberg, T.: Constraint integer programming. Ph.D. thesis, Technische Universität Berlin (2007)

    Google Scholar 

  2. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Oper. Res. Lett. 34(4), 1–12 (2006). doi:10.1016/j.orl.2005.07.009

    Article  MathSciNet  Google Scholar 

  4. Achterberg, T., Raack, C.: The MCF-separator: detecting and exploiting multi-commodity flow structures in MIPs. Math. Program. Comput. 2(2), 125–165 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Achterberg, T., Wunderling, R.: Mixed integer programming: analyzing 12 years of progress. In: Facets of Combinatorial Optimization, pp. 449–481. Springer (2013)

    Google Scholar 

  6. Berthold, T.: Primal heuristics for mixed integer programs. Diploma thesis, Technische Universität Berlin (2006)

    Google Scholar 

  7. Berthold, T.: Measuring the impact of primal heuristics. Oper. Res. Lett. 41(6), 611–614 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berthold, T.: Heuristic algorithms in global MINLP solvers. Ph.D. thesis, Technische Universität Berlin (2014)

    Google Scholar 

  9. Berthold, T.: RENS—the optimal rounding. Math. Program. Comput. 6(1), 33–54 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Berthold, T., Hendel, G.: Shift-and-propagate. J. Heuristics 21(1), 73–106 (2015)

    Google Scholar 

  11. Bixby, R.E.: A brief history of linear and mixed-integer programming computation. Documenta Mathematica pp. 107–121 (2012)

    Google Scholar 

  12. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)

    Google Scholar 

  13. Borndörfer, R., Grötschel, M., Jäger, U.: Planning problems in public transit. In: Grötschel, M., Lucas, K., Mehrmann, V. (eds.) Production Factor Mathematics, pp. 95–121. Springer, Berlin (2010)

    Chapter  Google Scholar 

  14. COR@L: MIP Instances (2014). http://coral.ie.lehigh.edu/data-sets/mixed-integer-instances/

  15. Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8(3), 250–255 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  16. Danna, E., Rothberg, E., Pape, C.L.: Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Program. 102(1), 71–90 (2004)

    Article  Google Scholar 

  17. Fischetti, M., Lodi, A.: Local branching. Math. Program. 98(1–3), 23–47 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fischetti, M., Lodi, A.: Heuristics in mixed integer programming. In: J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh, J.C. Smith (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley (2010)

    Google Scholar 

  19. Ghosh, S.: DINS, a MIP improvement heuristic. In: Fischetti, M., Williamson, D.P. (eds.) 12th International IPCO Conference, Proceedings of the Integer Programming and Combinatorial Optimization, LNCS, vol. 4513, pp. 310–323. Springer, Berlin (2007)

    Google Scholar 

  20. Heinz, S., Ku, W.Y., Beck, J.: Recent improvements using constraint integer programming for resource allocation and scheduling. In: Gomes, C., Sellmann, M. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. Lecture Notes in Computer Science, vol. 7874, pp. 12–27. Springer, Berlin (2013)

    Chapter  Google Scholar 

  21. Johnson, E.L., Padberg, M.W.: Degree-two inequalities, clique facets, and biperfect graphs. North-Holland Math. Stud. 66, 169–187 (1982)

    Article  MathSciNet  Google Scholar 

  22. Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math. Program. Comput. 3(2), 103–163 (2011)

    Article  MathSciNet  Google Scholar 

  23. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28(3), 497–520 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lee, E., Lewis, D.: Integer programming for telecommunications. In: Resende, M., Pardalos, P. (eds.) Handbook of Optimization in Telecommunications, pp. 67–102. Springer, US (2006)

    Chapter  Google Scholar 

  25. Lodi, A.: Mixed integer programming computation. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 619–645. Springer, Berlin (2010)

    Google Scholar 

  26. Lodi, A.: The heuristic (dark) side of MIP solvers. In: Talbi, E.G. (ed.) Hybrid Metaheuristics, Studies in Computational Intelligence, vol. 434, pp. 273–284. Springer, Berlin (2013)

    Chapter  Google Scholar 

  27. Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49(3), 363–371 (2001). doi:10.1287/opre.49.3.363.11211

    Article  MATH  MathSciNet  Google Scholar 

  28. Pochet, Y., Wolsey, L.A.: Production Planning by Mixed Integer Programming. Springer Science and Business Media, Heidelberg (2006)

    Google Scholar 

  29. Pryor, J., Chinneck, J.W.: Faster integer-feasibility in mixed-integer linear programs by branching to force change. Comput. Oper. Res. 38(8), 1143–1152 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rothberg, E.: An evolutionary algorithm for polishing mixed integer programming solutions. INFORMS J. Comput. 19(4), 534–541 (2007)

    Article  MATH  Google Scholar 

  31. Salvagnin, D.: Detecting and exploiting permutation structures in MIPs. In: Simonis, H. (ed.) Integration of AI and OR Techniques in Constraint Programming. Lecture Notes in Computer Science, vol. 8451, pp. 29–44. Springer, Berlin (2014)

    Chapter  Google Scholar 

  32. Savelsbergh, M.W.P.: Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Winkler, M.: Presolving for pseudo-Boolean optimization problems. Diploma thesis, Tech-nische Universität Berlin (2014)

    Google Scholar 

  34. Wunderling, R.: Paralleler und objektorientierter Simplex-algorithmus. Ph.D. thesis, Tech-nische Universität Berlin (1996)

    Google Scholar 

Download references

Acknowledgments

The work for this article has been conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (fund number 05M14ZAM). The authors would like to thank the anonymous reviewer for helpful comments on the paper.

Author information

Authors and Affiliations

  1. Zuse Institute Berlin, Takustr. 7, 14195, Berlin, Germany

    Gerald Gamrath

  2. Fair Isaac Europe Ltd, c/o ZIB, Takustr. 7, 14195, Berlin, Germany

    Timo Berthold & Stefan Heinz

  3. Gurobi GmbH, c/o ZIB, Takustr. 7, 14195, Berlin, Germany

    Michael Winkler

Authors
  1. Gerald Gamrath
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Timo Berthold
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Heinz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michael Winkler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerald Gamrath .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Katsuki Fujisawa

  2. Zuse Institute Berlin, Berlin, Germany

    Yuji Shinano

  3. Kyushu University, Fukuoka, Japan

    Hayato Waki

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Japan

About this paper

Cite this paper

Gamrath, G., Berthold, T., Heinz, S., Winkler, M. (2016). Structure-Based Primal Heuristics for Mixed Integer Programming. In: Fujisawa, K., Shinano, Y., Waki, H. (eds) Optimization in the Real World. Mathematics for Industry, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55420-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-55420-2_3

  • Published:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-55419-6

  • Online ISBN: 978-4-431-55420-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation