Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Information-based branching schemes for binary linear mixed integer problems

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

Branching variable selection can greatly affect the effectiveness and efficiency of a branch-and-bound algorithm. Traditional approaches to branching variable selection rely on estimating the effect of the candidate variables on the objective function. We propose an approach which is empowered by exploiting the information contained in a family of fathomed subproblems, collected beforehand from an incomplete branch-and-bound tree. In particular, we use this information to define new branching rules that reduce the risk of incurring inappropriate branchings. We provide computational results that demonstrate the effectiveness of the new branching rules on various benchmark instances.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achterberg T.: Conflict analysis in mixed integer programming. Discrete. Optim. 4, 4–20 (2007a)

    Article  MATH  MathSciNet  Google Scholar 

  2. Achterberg, T.: Constraint integer programming. Ph.D. thesis, Technische Universität Berlin. http://opus.kobv.de/tuberlin/volltexte/2007/1611/ (2007b)

  3. Achterberg T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1, 1–32 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Achterberg T., Berthold T.: Hybrid branching. In: van Hoeve, W.J., Hooker, J.N. (eds) CPAIOR, Lecture Notes in Computer Science, vol. 5547, pp. 309–311. Springer, Berlin (2009)

    Google Scholar 

  5. Achterberg T., Koch T., Martin A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Achterberg T., Koch T., Martin A.: Miplib 2003. Oper. Res. Lett. 34, 361–372 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Amaldi E., Pfetsch M.E., Trotter L.E.: On the maximum feasible subsystem problem, IISs, and IIS-hypergraphs. Math. Program. 95, 533–554 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Applegate, D., Bixby, R., Chvatal, V., Cook, W.: On the solution of traveling salesman problems. Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung. International Congress of Mathematicians, 645–656 (1998)

  9. Avenali A.: Resolution branch and bound and an application: the maximum weighted stable set problem. Oper. Res. 55, 932–948 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables. In: Proceedings of the Fifth International Conference on Operational Research, vol. 55, pp. 447–454. Tavistock publication, London, UK (1970)

  11. Bénichou M., Gauthier J.M., Girodet P., Hentges G., Ribière G., Vincen O.: Experiments in mixed-integer linear programming. Math. Program. 1, 76–94 (1971)

    Article  MATH  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. Chvátal V.: Resolution search. Discrete Appl. Math. 73, 81–99 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cor@l: Computational optimization research at lehigh university. http://coral.ie.lehigh.edu/mip-instances/ (2009)

  15. Cornuéjols, G., Liberti, L., Nannicini, G.: Improved strategies for branching on general disjunction. Optimization online, http://www.optimization-online.org/DB_FILE/2008/08/2071.pdf (2008)

  16. CPLEX. 11.1. ILOG. http://www.ilog.com/products/cplex

  17. Davey B., Boland N., Stuckey P.J.: Efficient intelligent backtracking using linear programming. INFORMS J. Comput. 14, 373–386 (2002)

    Article  MathSciNet  Google Scholar 

  18. Dixon H.E., Ginsberg M.L.: Combining satisfiability techniques from AI and OR. Knowl. Eng. Rev. 15, 31–45 (2000)

    Article  Google Scholar 

  19. Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Forrest J.J.H., Hirst J.P.H., Tomlin J.A.: Practical solution of large scale mixed integer programming problems with UMPIRE. Manag. Sci. 20, 736–773 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gilpin, A., Sandholm, T.: Information-theoretic approaches to branching in search. In: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI). Hyderabad, India (2007)

  22. Glankwamdee, W., Linderoth, J.T.: Lookahead branching for mixed integer programming. Optimization online, http://www.optimization-online.org/DB_FILE/2006/10/1490.pdf (2006)

  23. Gleeson J., Ryan J.: Identifying minimally infeasible subsystem of inequalities. ORSA J. Comput. 2, 61–63 (1990)

    MATH  Google Scholar 

  24. Gomes, C., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: Handbook of Knowledge Representation, Foundations of Artificial Intelligence, vol. 3, pp. 89–134. Elsevier, Amsterdam (2008)

  25. Hooker J.N.: Constraint satisfaction methods for generating valid cuts. In: Woodruff, D.L. (eds) Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search: Interfaces in Computer Science and Operations Research, pp. 1–30. Kluwer, Norwell (1998)

    Google Scholar 

  26. Hooker J.N.: Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley, New York (2000)

    MATH  Google Scholar 

  27. Karamanov, M., Cornuéjols, G.: Branching on general disjunctions. Technical report, http://integer.tepper.cmu.edu/webpub/d-branching.pdf (2005)

  28. Kılınç Karzan, F., Nemhauser, G.L., Savelsbergh, M.W.P.: Information based branching rules in integer programming. INFORMS Annual Meeting. Washington, DC, USA (2008)

  29. Linderoth J.T., Savelsbergh M.W.P.: A computational study of search strategies for mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mahajan, A., Ralphs, T.K.: Experiments with branching using general disjunctions. Optimization online, http://www.optimization-online.org/DB_FILE/2008/06/2020.pdf (2008a)

  31. Mahajan, A., Ralphs, T.K.: On the complexity of branching on general hyperplanes for integer programming. Optimization online, http://www.optimization-online.org/DB_FILE/2008/10/2119.pdf (2008b)

  32. Marques-Silva J.P., Sakallah K.A.: Grasp: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48, 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  33. MINTO. 3.1. MINTO: Mixed INTeger Optimizer. Georgia Institute of Technology, http://coral.ie.lehigh.edu/minto/

  34. Mittelmann, H.D.: Milptestset. http://plato.asu.edu/ftp/milp/ (2009)

  35. Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient sat solver. In: DAC’01: Proceedings of the 38th Annual Design Automation Conference (2001)

  36. Owen J.H., Mehrotra S.: Experimental results on using general disjunctions in branch-and-bound for general-integer linear programs. Comput. Optim. Appl. 20, 159–170 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Patel J., Chinneck J.W.: Active-constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Program. 110, 445–474 (2007)

    Article  MathSciNet  Google Scholar 

  38. Sandholm, T., Shields, R.: Nogood learning for mixed integer programming. Technical report, CMU-CS-06-155, Carnegie Mellon University, Computer Science Department, http://www.cs.cmu.edu/sandholm/nogoodsForMip.techReport06.pdf (2006)

  39. SCIP. 1.1.0. SCIP: solving constraint integer programs. Zuse Institute Berlin, http://scip.zib.de

  40. Stallman R.M., Sussman G.J.: Forward reasoning and dependency-directed backtracking in a system for computer-aided circuit analysis. Artif. Intell. 9, 135–196 (1977)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, NW, Atlanta, GA, 30332-0205, USA

    Fatma Kılınç Karzan, George L. Nemhauser & Martin W. P. Savelsbergh

Authors
  1. Fatma Kılınç Karzan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George L. Nemhauser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin W. P. Savelsbergh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fatma Kılınç Karzan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kılınç Karzan, F., Nemhauser, G.L. & Savelsbergh, M.W.P. Information-based branching schemes for binary linear mixed integer problems. Math. Prog. Comp. 1, 249–293 (2009). https://doi.org/10.1007/s12532-009-0009-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-009-0009-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation