Skip to main content
Springer Nature Link
Log in

Sparse Multi-term Disjunctive Cuts for the Epigraph of a Function of Binary Variables

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13265))

Abstract

We propose a new method for generating cuts valid for the epigraph of a function of binary variables. The proposed cuts are disjunctive cuts defined by many disjunctive terms obtained by enumerating a subset I of the binary variables. We show that by restricting the support of the cut to the same set of variables I, a cut can be obtained by solving a linear program with \(2^{|I|}\) constraints. While this limits the size of the set I used to define the multi-term disjunction, the procedure enables generation of multi-term disjunctive cuts using far more terms than existing approaches. Experience on three MILP problems with block diagonal structure using |I| up to size 10 indicates the sparse cuts can often close nearly as much gap as the multi-term disjunctive cuts without this restriction and in a fraction of the time.

This research is supported by the Office of Naval Research under grant N00014-21-1-2574.

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

Access this chapter

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

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, S.: Two-stage stochastic integer programming: a brief introduction. Wiley encyclopedia of operations research and management science (2010)

    Google Scholar 

  2. Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.: Cutting planes from two rows of a simplex tableau. In: Proceedings of IPCO XII, vol. 4513, pp. 1–15 (2007)

    Google Scholar 

  3. Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.: The mixed vertex packing problem. Math. Program. 89(1), 35–53 (2000)

    Article  MathSciNet  Google Scholar 

  4. Balas, E.: Disjunctive programming. In: Annals of Discrete Mathematics, vol. 5, pp. 3–51. Elsevier (1979)

    Google Scholar 

  5. Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discret. Appl. Math. 89(1–3), 3–44 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42(9), 1229–1246 (1996)

    Article  Google Scholar 

  8. Balas, E., Perregaard, M.: Lift-and-project for mixed 0-1 programming: recent progress. Discret. Appl. Math. 123(1–3), 129–154 (2002)

    Article  MathSciNet  Google Scholar 

  9. Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35(3), 704–720 (2010)

    Article  MathSciNet  Google Scholar 

  10. Bertsimas, D., Cory-Wright, R., Pauphilet, J.: A unified approach to mixed-integer optimization problems with logical constraints. SIAM J. Optim. 31(3), 2340–2367 (2021)

    Article  MathSciNet  Google Scholar 

  11. Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization, vol. 6. Athena Scientific, Belmont (1997)

    Google Scholar 

  12. Bodur, M., Dash, S., Günlük, O., Luedtke, J.: Strengthened Benders cuts for stochastic integer programs with continuous recourse. INFORMS J. Comput. 29(1), 77–91 (2017)

    Article  MathSciNet  Google Scholar 

  13. Ceria, S., Pataki, G.: Solving integer and disjunctive programs by lift and project. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 271–283. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-69346-7_21

    Chapter  MATH  Google Scholar 

  14. Chen, R.: Valid inequalities and integer programming formulations for stochastic integer programming and some machine learning problems. Ph.D. thesis, The University of Wisconsin-Madison (2021)

    Google Scholar 

  15. Chen, R., Luedtke, J.: On generating Lagrangian cuts for two-stage stochastic integer programs. arXiv preprint arXiv:2106.04023 (2021)

  16. Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47(1), 155–174 (1990)

    Article  Google Scholar 

  17. Cornuejols, G., Fisher, M.L., Nemhauser, G.L.: Exceptional paper-location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Manag. Sci. 23(8), 789–810 (1977)

    Article  Google Scholar 

  18. Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120(2), 429–456 (2009)

    Article  MathSciNet  Google Scholar 

  19. Cplex, IBM ILOG: V12. 1: User’s Manual for CPLEX (2009)

    Google Scholar 

  20. Dey, S.S., Iroume, A., Molinaro, M.: Some lower bounds on sparse outer approximations of polytopes. Oper. Res. Lett. 43(3), 323–328 (2015)

    Article  MathSciNet  Google Scholar 

  21. Dey, S.S., Lodi, A., Tramontani, A., Wolsey, L.A.: Experiments with two row tableau cuts. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 424–437. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13036-6_32

    Chapter  Google Scholar 

  22. Dey, S.S., Molinaro, M., Wang, Q.: Approximating polyhedra with sparse inequalities. Math. Program. 154(1), 329–352 (2015)

    Article  MathSciNet  Google Scholar 

  23. Dey, S.S., Molinaro, M., Wang, Q.: Analysis of sparse cutting planes for sparse MILPs with applications to stochastic MILPs. Math. Oper. Res. 43(1), 304–332 (2018)

    Article  MathSciNet  Google Scholar 

  24. Fukasawa, R., Poirrier, L., Yang, S.: Split cuts from sparse disjunctions. Math. Program. Comput. 12(2), 295–335 (2020). https://doi.org/10.1007/s12532-020-00180-9

    Article  MathSciNet  MATH  Google Scholar 

  25. Gade, D., Küçükyavuz, S., Sen, S.: Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs. Math. Program. 144(1–2), 39–64 (2014)

    Article  MathSciNet  Google Scholar 

  26. Gray, F.: Pulse code communication. United States Patent Number 2632058 (1953)

    Google Scholar 

  27. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  28. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90(3), 429–457 (2001)

    Article  MathSciNet  Google Scholar 

  29. Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (2021). https://www.gurobi.com

  30. Laporte, G., Louveaux, F.V.: The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 13(3), 133–142 (1993)

    Article  MathSciNet  Google Scholar 

  31. Li, Y., Richard, J.P.P.: Cook, Kannan and Schrijver’s example revisited. Discret. Optim. 5(4), 724–734 (2008)

    Article  MathSciNet  Google Scholar 

  32. Méndez-Díaz, I., Miranda-Bront, J.J., Vulcano, G., Zabala, P.: A branch-and-cut algorithm for the latent-class logit assortment problem. Discret. Appl. Math. 164, 246–263 (2014)

    Article  MathSciNet  Google Scholar 

  33. Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions-I. Math. Program. 14(1), 265–294 (1978)

    Article  MathSciNet  Google Scholar 

  34. Ntaimo, L.: Fenchel decomposition for stochastic mixed-integer programming. J. Global Optim. 55(1), 141–163 (2013)

    Article  MathSciNet  Google Scholar 

  35. Pan, F., Morton, D.P.: Minimizing a stochastic maximum-reliability path. Networks 52(3), 111–119 (2008)

    Article  MathSciNet  Google Scholar 

  36. Perregaard, M., Balas, E.: Generating cuts from multiple-term disjunctions. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 348–360. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45535-3_27

    Chapter  Google Scholar 

  37. Rahmaniani, R., Ahmed, S., Crainic, T.G., Gendreau, M., Rei, W.: The Benders dual decomposition method. Oper. Res. 68, 878–895 (2020)

    Article  MathSciNet  Google Scholar 

  38. Sen, S., Higle, J.L.: The C3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: set convexification. Math. Program. 104(1), 1–20 (2005)

    Article  MathSciNet  Google Scholar 

  39. Sen, S., Sherali, H.D.: Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math. Program. 106(2), 203–223 (2006)

    Article  MathSciNet  Google Scholar 

  40. Wiegele, A.: Biq Mac Library-a collection of max-cut and quadratic 0-1 programming instances of medium size. Preprint 51 (2007)

    Google Scholar 

  41. Zhang, M., Küçükyavuz, S.: Finitely convergent decomposition algorithms for two-stage stochastic pure integer programs. SIAM J. Optim. 24(4), 1933–1951 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cornell Tech, New York, NY, USA

    Rui Chen

  2. University of Wisconsin-Madison, Madison, WI, USA

    James Luedtke

Authors
  1. Rui Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Luedtke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Chen .

Editor information

Editors and Affiliations

  1. Delft University of Technology, Delft, The Netherlands

    Karen Aardal

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Laura Sanità

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chen, R., Luedtke, J. (2022). Sparse Multi-term Disjunctive Cuts for the Epigraph of a Function of Binary Variables. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-06901-7_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-06900-0

  • Online ISBN: 978-3-031-06901-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics