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

Advertisement

Springer Nature Link
Log in

On the facets of the mixed–integer knapsack polyhedron

  • Published:
Mathematical Programming Submit manuscript

Abstract.

We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.

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. Atamtürk, A.: Sequence independent lifting for mixed–integer programming. Nov 2001. Available at http://ieor.berkeley.edu/∼atamturk.

  2. Atamtürk, A.: On capacitated network design cut–set polyhedra. Math. Program. 92, 425–437 (2002)

  3. Atamtürk, A., Rajan, D.: On splittable and unsplittable flow capacitated network design arc–set polyhedra. Math. Program. 92, 315–333 (2002)

  4. Bachem, A., Johnson, E.L., Schrader, R.: A characterization of minimal valid inequalities for mixed integer programs. Oper. Res. Lett. 1, 63–66 (1982)

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

  6. Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.R.: Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)

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

  8. Bienstock, D., Günlük, O.: Capacitated network design – Polyhedral structure and computation. INFORMS J. Comput. 8, 243–259 (1996)

  9. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 54, (1998) Available from URL http://www.caam.rice.edu/∼bixby/miplib/miplib.html.

  10. Brockmüller, B., Günlük, O., Wolsey, L.A.: Designing private line networks – Polyhedral analysis and computation. CORE Discussion Paper 9647, Université Catholique de Louvain, 1996

  11. Ceria, S., Cordier, C., Marchand, H., Wolsey, L.A.: Cutting planes for integer programs with general integer variables. Math. Program. 81, 201–214 (1998)

  12. Cornuéjols, G., Li, Y., Vandenbussche, D.: k-cuts: A variation of Gomory mixed integer cuts from the LP tableau. Technical report, Feb 2000. To appear in INFORMS Journal on Computing

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

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

  15. Gomory, R.E.: On the relation between integer and non-integer solutions to linear programs. Proceeding of the National Academy of Science 53, 260–265 (1965)

  16. 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)

  17. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Lifted flow cover inequalities for mixed 0–1 integer programs. Math. Program. 85, 439–467 (1999)

  18. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Sequence independent lifting in mixed integer programming. J. Combin. Optim. 4, 109–129 (2000)

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

  20. Magnanti, T.L., Mirchandani, P.: Shortest paths, single origin-destination network design, and associated polyhedra. Networks 23, 103–121 (1993)

  21. Magnanti, T.L., Mirchandani, P., Vachani, R.: The convex hull of two core capacitated network design problems. Math. Program. 60, 233–250 (1993)

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

  23. Marchand, H., Wolsey, L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)

  24. Martin, A., Weismantel, R.: Contribution to general mixed integer knapsack problems. Technical Report SC 97-38, Konrad–Zuse–Zentrum für Informationstechnik Berlin, 1997

  25. Miller, A., Wolsey, L.A.: Discrete lot–sizing and convex integer programming. CORE Discussion Paper 0108, Université Catholique de Louvain, 2001

  26. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley and Sons, New York, 1988

  27. 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)

  28. Padberg, M.W.: Covering, packing and knapsack problems. Annals of Discre. Math. 4, 265–287 (1979)

  29. Pochet, Y., Weismantel, R.: The sequential knapsack polytope. SIAM J. Optim. 8, 248–264 (1998)

  30. Pochet, Y., Wolsey, L.A.: Integer knapsack and flow covers with divisible coefficients: Polyhedra, optimization, and separation. Discre. Appl. Math. 59, 57–74 (1995)

  31. Richard, J.-P.P., de Farias, I.R., Nemhauser, G.L.: Lifted inequalities for 0–1 mixed integer programming : Basic theory and algorithms. In: Integer Programming and Combinatorial Optimization, 9th International IPCO Conference, Proceedings, pages 161–175, 2002

  32. Weismantel, R.: On the 0/1 knapsack polytope. Math. Program. 77, 49–68 (1997)

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

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

  35. Wolsey, L.A.: Valid inequalities and superadditivity for 0/1 integer programs. Math. Oper. Res. 2, 66–77 (1977)

  36. Wolsey, L.A.: Integer Programming. John Wiley and Sons, New York, 1998

  37. Zemel, E.: Easily computable facets of the knapsack polytope. Math. Oper. Res. 14, 760–764 (1989)

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, University of California at Berkeley, Berkeley, CA, 94720–1777

    Alper Atamtürk

Authors
  1. Alper Atamtürk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alper Atamtürk.

Additional information

Supported, in part, by NSF grants DMII–0070127 and DMII–0218265.

Mathematics Subject Classification (2000): 90C10, 90C11, 90C57

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atamtürk, A. On the facets of the mixed–integer knapsack polyhedron. Math. Program., Ser. B 98, 145–175 (2003). https://doi.org/10.1007/s10107-003-0400-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0400-z

Keywords

Search

Navigation