Skip to main content
SpringerLink
Log in

A Unified Framework for Modeling and Solving Combinatorial Optimization Problems: A Tutorial

  • Chapter
Multiscale Optimization Methods and Applications

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 82))

Summary

In recent years the unconstrained quadratic binary program (UQP) has emerged as a unified framework for modeling and solving a wide variety of combinatorial optimization problems. This tutorial gives an introduction to this evolving area. The methodology is illustrated by several examples and substantial computational experience demonstrating the viability and robustness of the approach.

Earlier versions of this material appear in references [KGAR04a, KGAR04b]

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alidaee, B, G. Kochenberger, and A. Ahmadian, “0–1 Quadratic Programming Approach for the Optimal Solution of Two Scheduling Problems,” International Journal of Systems Science, 25, 401–408, 1994.

    MathSciNet  MATH  Google Scholar 

  2. Alkhamis, T. M., M. Hasan, and M. A. Ahmed, “Simulated Annealing for the Unconstrained Binary Quadratic Pseudo-Boolean Function,” European Journal of Operational Research, 108, (1998), 641–652.

    Article  MATH  Google Scholar 

  3. Amini, M., B. Alidaee, G. Kochenberger, “A Scatter Search Approach to Unconstrained Quadratic Binary Programs,” New Methods in Optimization, Cone, Dorigo, and Glover, eds., McGraw-Hill, 317–330, 1999.

    Google Scholar 

  4. Beasley, J. E., “Heuristic Algorithms for the Unconstrained Binary Quadratic Programming Problem, Working Paper, Imperial College, 1999.

    Google Scholar 

  5. Billionet, A. and A. Sutter, “Minimization of a Quadratic Pseudo-Boolean Function.” European Journal of OR, 78 pp. 106–115, (1994).

    Article  Google Scholar 

  6. Boros, E. and P. Hammer, “Pseudo-Boolean Optimization,” Discrete Applied Mathematics, 123(1–3) 155–225 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Boros, E., and P. Hammer, “The Max-Cut Problem and Quadratic 0–1 Optimization, Polyhedral Aspects, Relaxations and Bounds,” Annals of OR, 33, 151–225 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Boros, E, P. Hammer, and X, Sun, “The DDT Method for Quadratic 0–1 Minimization,” RUTCOR Research Center, RRR 39–89, 1989.

    Google Scholar 

  9. Boros, E. and A. Prekopa, “Probabilistic Bounds and Algorithms for the Maximum Satisfiability Problem,” Annals of OR, 21 (1989), pp. 109–126.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bourjolly, J.M., P. Gill, G. Laporte, and H. Mercure, “A Quadratic 0/1 Optimization Algorithm for the Maximum Clique and Stable Set Problems,” Working paper, University of Montreal, (1994).

    Google Scholar 

  11. Chardaire, P, and A. Sutter, “A Decomposition Method for Quadratic Zero-One Programming,” Management Science, 41:4, 704–712, 1994.

    Google Scholar 

  12. Cornuejolos, G., Combinatorial Optimization: Packing and Covering, CBMS-NSF, SIAM, (2001).

    Google Scholar 

  13. De Simone, C. M. Diehl, M. Junger, P. Mutzel, G. Reinelt, and G. Rinaldi, ‘Exact Ground State4s of Ising Spin Glasses: New Experimental Results with a Branch and Cut Algorithm,” Journal of Statistical Physics, 80, 487–496 (1995)

    Article  MATH  Google Scholar 

  14. Delorme, X., X. Gandibleau, and J. Rodriques, “GRASP for Set packing,” EJOR, 153, (2004), pp. 564–580.

    Article  MATH  Google Scholar 

  15. Glover, F., B. Alidaee, C. Rego, and G. Kochenberger, “One-Pass Heuristics for Large-Scale Unconstrained Binary Quadratic Programs,” EJOR 137, pp. 272–287, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  16. Glover, F, and M. Laguna, “Tabu Search,” Kluwer Academic Publishers, 1997.

    Google Scholar 

  17. Glover, F., G. Kochenberger, B. Alidaee, and M.M. Amini, “Tabu with Search Critical Event Memory: An Enhanced Application for Binary Quadratic Programs,” In: MetaHeuristics: Advances and Trends in Local Search Paradigms for Optimization, (Eds.) S. Voss, S. Martello, I. Osman, and C. Roucairol. Kluwer Academic Publisher, Boston, 1999.

    Google Scholar 

  18. Glover, F., G. Kochenberger., and B. Alidaee, “Adaptive Memory Tabu Search for Binary Quadratic Programs,” Management Science, 44:3, 336–345, 1998.

    MATH  Google Scholar 

  19. Grotschel, M., M. Junger, and G. Reinelt, “An Application of Combinatorial Optimization to Statistical Physics and Circuit Layout Design,” Operations Research, Vol.36, #3, May–June, (1988), pp. 493–513.

    Google Scholar 

  20. Hammer, P., and S. Rudeanu, Boolean Methods in Operations Research, Springer-Verlag, New York, 1968.

    MATH  Google Scholar 

  21. Hansen, P.B., “Methods of Nonlinear 0–1 Programming,” Annals Discrete Math, vol. 5, pp.53–70, 1979.

    Article  MATH  Google Scholar 

  22. Hansen, P, and B. Jaumard, “Algorithms for the Maximum Satisfiability Problem,” Computing, 44, 279–303 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  23. Hansen, P, B. Jaumard., and V. Mathon, “Constrained Nonlinear 0–1 Programming,” INFORMS Journal on Computing, 5:2, 97–119, 1993.

    MathSciNet  MATH  Google Scholar 

  24. Iasemidis, L. D., D. S. Shiau, J.C. Sackellares, and P. Pardalos, “Transition to Epileptic Seizures: Optimization,” DIMACS Series in Discrete Math and Theoretical Computer Science, Vol. 55, (2000), pp. 55–73.

    MathSciNet  Google Scholar 

  25. Joseph, A. “A Concurrent Processing Framework for the Set Partitioning Problem,” Computers & Operations Research, 29, 1375–1391, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  26. Katayama, K., M. Tani, and H. Narihisa, “Solving Large Binary Quadratic Programming Problems by an Effective Genetic Local Search Algorithm,” In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’00). Morgan Kaufmann, 2000.

    Google Scholar 

  27. Kochenberger, G., F. Glover, B. Alidaee, and C. Rego, “A Unified Modeling and Solution Framework for Combinatorial Optimization Problems,” OR Spectrum, 26, pp. 237–250, (2004).

    Article  MATH  Google Scholar 

  28. Kochenberger, G., F. Glover, B. Alidaee, and C. Rego, “Solving Combinatorial Optimization Problems via Reformulation and Adaptive Memory Metaheuristics,” Revolutionary Visions In Evolutionary Computation, ed. Anil Menon and David Goldberg, Kluwer Publisher, (to appear in 2004)

    Google Scholar 

  29. Kochenberger, G., F. Glover, B. Alidaee, and C. Rego, “An Unconstrained Quadratic Binary Programming Approach to the Vertex Coloring Problem,” Working Paper, University of Colorado at Denver, 2002.

    Google Scholar 

  30. Laughunn, D.J, “Quadratic Binary programming,” Operations Research, 14, 454–461, 1970.

    Google Scholar 

  31. Lewis, M., B. Alidaee, and G. Kochenberger, “Using xQx to Model and Solve the Uncapacitated Task Allocation Problem,” To Appear in OR Letters (2004).

    Google Scholar 

  32. Lodi, A., K. Allemand, and T. M. Liebling, “An Evolutionary Heuristic for Quadratic 0–1 Programming,” Technical Report OR-97-12, D.E.I.S., University of Bologna, 1997.

    Google Scholar 

  33. Merz, P. and B. Freisleben, “Genetic Algorithms for Binary Quadratic Programming,” Proceedings of the 1999 International Genetic and Evolutionary Computation Conference (GECCO’99), pp. 417–424, Morgan Kaufmann, 1999.

    Google Scholar 

  34. Mingozzi, A., M. Boschetti, S. Ricciardelli and L. Blanco, “A Set Partitioning Approach to the Crew Scheduling Problem,” Operations Research, 47(6) 873–888, 1999.

    Article  MATH  Google Scholar 

  35. Padberg, M., “On the Facial Structure of set packing Polyhedra,” Mathematical Programmming, vol 5, (1973) pp.199–215.

    Article  MATH  MathSciNet  Google Scholar 

  36. Palubeckis, G. “A Heuristic-Branch and Bound Algorithm for the Unconstrained Quadratic Zero-One Programming Problem,” Computing, pp. 284–301, (1995).

    Google Scholar 

  37. Pardalos, P, and G.P. Rodgers, “Computational Aspects of a Branch and Bound Algorithm for Quadratic Zero-one Programming,” Computing, 45, 131–144, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  38. Pardalos, P, and G.P. Rodgers, “A Branch and Bound Algorithm for Maximum Clique problem,” Computers & OR, 19, 363–375, 1992.

    Article  MATH  Google Scholar 

  39. Pardalos, P, and J. Xue, “The Maximum Clique Problem,” The Journal of Global Optimization, 4, 301–328, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  40. Pekec, A., and M. Rothkopf, “Combinatorial Auction Design,” Management Science, Vol. 49, #11, Nov. 2003, pp. 1485–1503.

    Article  Google Scholar 

  41. Ronnqvist, M., “Optimization in Forestry,” Math. Programming, Series B, 97, (2003). Pp. 267–284.

    MathSciNet  Google Scholar 

  42. Schrage, L., Optimization Modeling with LINDO, Duxbury Press, (1997)

    Google Scholar 

  43. Vemuganti, R, “Applications of Set Covering, Set Packing and Set Partitioning Models: A Survey,” Handbook of Combinatorial Optimization, eds. D. Zhu and P. Pardalos, Kluwer Academic Publishers, (1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Business, University of Colorado, Denver, Colorado, 80217, USA

    Gary A. Kochenberger

  2. School of Business, University of Colorado, Boulder, Colorado, 80304, USA

    Fred Glover

Authors
  1. Gary A. Kochenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fred Glover
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Florida, Gainesville, Florida

    William W. Hager , Shu-Jen Huang , Panos M. Pardalos  & Oleg A. Prokopyev , ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Kochenberger, G.A., Glover, F. (2006). A Unified Framework for Modeling and Solving Combinatorial Optimization Problems: A Tutorial. In: Hager, W.W., Huang, SJ., Pardalos, P.M., Prokopyev, O.A. (eds) Multiscale Optimization Methods and Applications. Nonconvex Optimization and Its Applications, vol 82. Springer, Boston, MA. https://doi.org/10.1007/0-387-29550-X_4

Download citation

Publish with us

Policies and ethics

Search

Navigation