Skip to main content
SpringerLink
Log in

An Exact Rational Mixed-Integer Programming Solver

  • Conference paper
Integer Programming and Combinatoral Optimization (IPCO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6655))

Abstract

We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the Miplib and Mittelmann collections.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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. Programming Computation 1(1), 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Achterberg, T., Koch, T., Martin, A.: The mixed integer programming library: MIPLIB (2003), http://miplib.zib.de

  4. Althaus, E., Dumitriu, D.: Fast and accurate bounds on linear programs. In: Vahrenhold, J. (ed.) SEA 2009. LNCS, vol. 5526, pp. 40–50. Springer, Heidelberg (2009)

    Google Scholar 

  5. Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  6. Applegate, D.L., Cook, W.J., Dash, S., Espinoza, D.G.: QSopt_ex, http://www.dii.uchile.cl/~daespino/ESolver_doc/main.html

  7. Applegate, D.L., Cook, W.J., Dash, S., Espinoza, D.G.: Exact solutions to linear programming problems. Oper. Res. Lett. 35(6), 693–699 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. Cook, W.J., Dash, S., Fukasawa, R., Goycoolea, M.: Numerically safe Gomory mixed-integer cuts. INFORMS J. Comput. 21(4), 641–649 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dhiflaoui, M., Funke, S., Kwappik, C., Mehlhorn, K., Seel, M., Schömer, E., Schulte, R., Weber, D.: Certifying and repairing solutions to large LPs, how good are LP-solvers? In: SODA 2003, pp. 255–256. ACM/SIAM (2003)

    Google Scholar 

  11. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Programming 91(2), 201–213 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Espinoza, D.G.: On Linear Programming, Integer Programming and Cutting Planes. Ph.D. thesis, Georgia Institute of Technology (2006)

    Google Scholar 

  13. GMP: GNU multiple precision arithmetic library, http://gmplib.org

  14. Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys (CSUR) 23(1), 5–48 (1991)

    Article  Google Scholar 

  15. IBM ILOG: CPLEX, http://www.ilog.com/products/cplex

  16. Koch, T.: The final NETLIB-LP results. Oper. Res. Lett. 32(2), 138–142 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kwappik, C.: Exact Linear Programming. Master thesis, Universität des Saarlandes (1998)

    Google Scholar 

  18. Mittelmann, H.D.: Benchmarks for Optimization Software (2010), http://plato.asu.edu/bench.html

  19. Neumaier, A., Shcherbina, O.: Safe bounds in linear and mixed-integer linear programming. Math. Programming 99(2), 283–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Steffy, D.E.: Topics in Exact Precision Mathematical Programming. Ph.D. thesis, Georgia Institute of Technology (2011)

    Google Scholar 

  21. de Vries, S., Vohra, R.: Combinatorial Auctions: A Survey. INFORMS J. Comput. 15(3), 284–309 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wilken, K., Liu, J., Heffernan, M.: Optimal instruction scheduling using integer programming. SIGPLAN Notices 35(5), 121–133 (2000)

    Article  Google Scholar 

  23. Zuse Institute Berlin: SCIP, http://scip.zib.de

Download references

Author information

Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    William Cook & Daniel E. Steffy

  2. Zuse Institute Berlin, Germany

    Thorsten Koch & Kati Wolter

Authors
  1. William Cook
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thorsten Koch
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel E. Steffy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kati Wolter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Oktay Günlük

  2. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600, Eindhoven, MB, The Netherlands

    Gerhard J. Woeginger

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cook, W., Koch, T., Steffy, D.E., Wolter, K. (2011). An Exact Rational Mixed-Integer Programming Solver. In: Günlük, O., Woeginger, G.J. (eds) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, vol 6655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20807-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20807-2_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20806-5

  • Online ISBN: 978-3-642-20807-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation