Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

  • Published:
Mathematical Programming Submit manuscript

Abstract.

We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.

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

Access this article

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.

Similar content being viewed by others

References

  1. Balas, E.: Facets of the knapsack polytope. Math. Program. 8, 146–164 (1975)

    Google Scholar 

  2. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58, 295–324 (1993)

    Google Scholar 

  3. Balas, E., Zemel, E.: Facets of the knapsack polytope from minimal covers. SIAM J. Appl. Math. 34, 119–148 (1978)

    Google Scholar 

  4. Christof, T., Løbel, A.: PORTA: a polyhedron representation transformation algorithm, http://www.zib.de/Optimization/Software/Porta/, 1997

  5. Crowder, H.P., Johnson, E.L., Padberg, M.W.: Solving large-scale zero-one linear programming problems. Operations Res. 31, 803–834 (1983)

    Google Scholar 

  6. de Farias, I.R.: A polyhedral approach to combinatorial complementarity programming problems. Ph.D. thesis, School of Industrial and Systems Engineering Georgia Institute of Technology, 1995

  7. de Farias, I.R., Johnson, E.L., Nemhauser, G.L.: A generalized assignment problem with special ordered sets: A Polyhedral Approach. Math. Program. 89, 187–203 (2000)

    Google Scholar 

  8. de Farias, I.R., Johnson, E.L., Nemhauser, G.L.: Facets of the complementarity knapsack polytope. Math. Operations Res. 27, 210–226 (2002)

    Google Scholar 

  9. de Farias, I.R., Nemhauser, G.L.: A family of inequalities for the generalized assignment polytope. Operations Res. Let. 29, 49–51 (2001)

    Google Scholar 

  10. de Farias, I.R., Nemhauser, G.L.: A polyhedral study of the cardinality constrained knapsack problem. Technical Report 01-05, Georgia Institute of Technology, 2001

  11. Gomory, R.E.: An algorithm for the mixed integer problem. Technical Report RM-2597, RAND Corporation, 1960

  12. Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra and Its Appl. 2, 451–558 (1969)

    Google Scholar 

  13. Gu, Z.: Lifted cover inequalities for 0-1 and mixed 0-1 integer programs. Ph.D. thesis, School of Industrial and Systems Engineering Georgia Institute of Technology, 1995

  14. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted flow cover inequalities for mixed 0-1 integer programs. Georgia institute of technology, 1996

  15. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted cover inequalities for 0-1 integer programs: Computation. INFORMS J. Comput. 10, 427–437 (1998)

    Google Scholar 

  16. Hammer, P.L., Johnson, E.L., Peled, U.N.: Facets of regular 0-1 polytopes. Math. Program. 8, 179–206 (1975)

    Google Scholar 

  17. Jünger, M., Reinelt, G., Rinaldi, G.: Combinatorial Optimization – Eureka, You Shrink!, Papers Dedicated to Jack Edmonds, 5th International Workshop, Aussois, France, March 5–9, 2001, Revised Papers, Combinatorial Optimizaiton, Springer, Lecture Notes in Computer Science, 2570, 3-540-00580-3, DBLP, http://dblp.uni-trier.de 2003

  18. Marchand, H., Wolsey, L.A.: The 0-1 knapsack problem with a single continuous variable. Math. Program. 85, 15–33 (1999)

    Google Scholar 

  19. Nemhauser, G.L., Wolsey, L.A.: A recursive procedure for generating all cuts for 0-1 mixed integer programs. Math. Program. 46, 379–390 (1990)

    Google Scholar 

  20. Padberg, M.W.: On the facial structure of set packing polyhedra. Math. Program. 5, 199–215 (1973)

    Google Scholar 

  21. Richard, J.-P.P.: Lifted inequalities for 0-1 mixed integer programming. Georgia Institute of Technology, 2002

  22. Wolsey, L.A.: Faces for a linear inequality in 0-1 variables. Math. Program. 8, 165–178 (1975)

    Google Scholar 

  23. Wolsey, L.A.: Facets and strong valid inequalities for integer programs. Operations Res. 24, 367–372 (1976)

    Google Scholar 

  24. Zemel, E.: Lifting the facets of zero-one polytopes. Math. Program. 15, 268–277 (1978)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J.-P.P. Richard.

Additional information

This research was supported by NSF grants DMI-0100020 and DMI-0121495

Mathematics Subject Classification (2000): 90C11, 90C27

Rights and permissions

Reprints and permissions

About this article

Cite this article

Richard, JP., de Farias Jr, I. & Nemhauser, G. Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms. Math. Program., Ser. B 98, 89–113 (2003). https://doi.org/10.1007/s10107-003-0398-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0398-2

Keywords