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

Advertisement

Springer Nature Link
Log in

A constraint generation algorithm for large scale linear programs using multiple-points separation

  • Published:
Mathematical Programming Submit manuscript

Abstract

In order to solve linear programs with a large number of constraints, constraint generation techniques are often used. In these algorithms, a relaxation of the formulation containing only a subset of the constraints is first solved. Then a separation procedure is called which adds to the relaxation any inequality of the formulation that is violated by the current solution. The process is iterated until no violated inequality can be found. In this paper, we present a separation procedure that uses several points to generate violated constraints. The complexity of this separation procedure and of some related problems is studied. Also, preliminary computational results about the advantages of using multiple-points separation procedures over traditional separation procedures are given for random linear programs and survivable network design. They illustrate that, for some specific families of linear programs, multiple-points separation can be computationally effective.

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. Astorino, A., Gaudioso, M.: Polyhedral separability through successive LP. J. Optim. Theory Appl. 112 (2), 265–293 (2002)

    Article  MathSciNet  Google Scholar 

  2. Ben-Ameur, W., Neto, J.: Acceleration of cutting plane and column generation algorithms. 2003 (Submitted)

  3. Ben-Ameur, W., Neto, J.: Multipoint separation. Research Report 04012 RS2M, Institut National des Télécommunications, Evry, France (2004)

  4. Benders, J.F.: Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bennett, K.P., Mangasarian, O.L.: Robust linear programming discrimination of two linearly inseparable sets. Optim. Meth. and Software 1, 23–34 (1992)

    Google Scholar 

  6. Bosch, R.A., Smith, J.A.: Separating hyperplanes and the authorship of the disputed federalist papers. Amer. Math. Monthly 105 (7), 601–608 (1998)

    Article  MathSciNet  Google Scholar 

  7. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, 2004

  8. Cheston, G.A., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: On the computational complexity of upper fractional domination. Discrete Appl. Math. 27, 195–207 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Disc. Math. 4, 305–337

  10. CPLEX, CPLEX callable library, CPLEX Optimization, Inc

  11. Dahl, G., Stoer, M.: A cutting plane algorithm for multicommodity survivable network design problems. INFORMS J. Comput. 10, 1–11 (1998)

    MathSciNet  Google Scholar 

  12. Dahl, G., Stoer, M.: A polyhedral approach to multicommodity survivable network design. Numerische Mathematik 68, 149–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and intractability. W.H. Freeman, New York, 1979

  14. Geoffrion, A.M.: Generalized Benders decomposition. J. Optim. Theory and Appl. 10, 237–260 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  15. Goffin, J.L., Vial, J.P.: Convex nondifferentiable optimization: a survey focussed on the analytic center cutting plane method. Optim. Meth. Software 17, 805–867 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64, 275–278 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  17. Grötschel, M., Lovasz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, 1988

  18. Iri, M.: On an extension of the maximum-flow minimum-cut theorem to multicommodity flows. J. Oper. Res. Soc. Japan 13, 129–135 (1971)

    MATH  MathSciNet  Google Scholar 

  19. Jünger, M., Reinelt, G., Thienel, S.: Practical problem solving with cutting plane algorithms in combinatorial optimization. In: Cook, W., Lovasz, L., Seymour, P (eds.) Combinatorial optimization, DIMACS series in discrete mathematics and theoretical computer science, 1995 pp 111–152

  20. Kelley, J.E.: The cutting-plane method for solving convex programs. J. SIAM 8, 11–152 (1960)

    MathSciNet  Google Scholar 

  21. Khachiyan, L.G.: A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20, 191–194 (1979)

    MATH  Google Scholar 

  22. Mangasarian, O.L.: Linear and nonlinear separation of patterns by linear programming. Oper. Res. 13 (3), 444–452 (1965)

    MathSciNet  Google Scholar 

  23. Mangasarian, O.L., Setiono, R., Wolberg, W.W.: Pattern recognition via linear programming: theory and applications to medical diagnosis. In: Coleman, T.F., Li, Y (eds) Large-scale numerical optimization. SIAM, Philadelphia, 1990 pp 22–30

  24. Mangasarian, O.L., Street, W.N., Wolberg, W.W.: Breast cancer diagnosis and prognosis via linear programming. Oper. Res. 43, 570–577 (1995)

    MATH  MathSciNet  Google Scholar 

  25. Megiddo, N.: On the complexity of polyhedral separability. Disc. Comput. Geom. 325–337 (1988)

  26. Minoux, M.: Network synthesis and optimum network design problems: Models, solution methods and applications. Networks, 19, 313–360 (1989)

    MATH  MathSciNet  Google Scholar 

  27. Mitchell, J.E.: Interior point methods for combinatorial optimization. In: Terlaky, T. (ed.) Interior point methods in mathematical programming, Kluwer Academic Publishers, 1996

  28. Mitchell, J.E., Borchers, B.: Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. Technical Report, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180–3590, (1997)

  29. Mitchell, J.E., Pardalos, P., Resende, M.G.C.: Interior point methods for combinatorial optimization. In: Du, D.-Z., Pardalos, P. (ed.) Handbook of combinatorial optimization Kluwer Academic Publishers, 1998

  30. Nemhauser, G.L., Wolsey, L.A.: Integer and combinatorial optimization. John Wiley, New York, 1988

  31. Nesterov, Y.: Cutting plane algorithms from analytic analytic centers: efficiency, estimates. Math. Prog. 56, 149–176 (1995)

    MathSciNet  Google Scholar 

  32. Onaga, K., Kakusho, O.: On feasibility conditions of multicommodity flows in networks. Trans. Circ. Theory 4, 425–429 (1971)

    MathSciNet  Google Scholar 

  33. Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev 33, 60–100 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  34. Rosen, J.B.: Pattern separation by convex programming. J. Math. Anal. Appl. 10, 123–134 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  35. Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. Math. Prog. 73, 291–341 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  36. Veinott, A.F.: The supporting hyperplane method for unimodal programming. Oper. res. 1, 147–152 (1967)

    Article  MathSciNet  Google Scholar 

  37. Ye, Y.: Interior-point algorithm: theory and analysis. John Wiley & Sons, 1997

Download references

Author information

Authors and Affiliations

  1. GET/INT, 9 rue Charles Fourier, 91011, Evry Cedex, France

    Walid Ben-Ameur & José Neto

Authors
  1. Walid Ben-Ameur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Neto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ben-Ameur, W., Neto, J. A constraint generation algorithm for large scale linear programs using multiple-points separation. Math. Program. 107, 517–537 (2006). https://doi.org/10.1007/s10107-005-0694-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0694-0

Keywords

Mathematics Subject classification (2000)

Search

Navigation