Branch-and-cut for complementarity-constrained optimization | Mathematical Programming Computation Skip to main content
Springer Nature Link
Log in

Branch-and-cut for complementarity-constrained optimization

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

We report and analyze the results of our computational testing of branch-and-cut for the complementarity-constrained optimization problem (CCOP). Besides the MIP cuts commonly present in commercial optimization software, we used inequalities that explore complementarity constraints. To do so, we generalized two families of cuts proposed earlier by de Farias, Johnson, and Nemhauser that had never been tested computationally. Our test problems consisted of linear, binary, and general integer programs with complementarity constraints. Our results on the use of complementarity cuts within a major commercial optimization solver show that they are of critical importance to tackling difficult CCOP instances, typically reducing the computational time required to solve them tremendously.

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
¥17,985 /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 includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables. In: Lawrence, J. (ed.) Proceedings of the Fifth International Conference on Operations Research, pp. 447–454. Tavistock Publications, London (1970)

  3. Bean, J., Noon, C., Ryan, S., Salton, G.: Selecting tenants in a shopping mall. Interfaces 18, 1–9 (1988)

    Article  Google Scholar 

  4. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Danna, E.: Solving optimization problems at google. Talk presented at the 2010 INFORMS national meeting, Austin (2010)

  7. Dantzig, G.B.: On the significance of solving linear programming problems with some integer variables. Econometrica 28, 30–44 (1960)

    Article  MATH  MathSciNet  Google Scholar 

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

  9. de Farias, JR, I.R., Kozyreff, E., Gupta, R., Zhao, R.: Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints. Math. Progr. C 5(1), 75–112 (2013)

  10. Dowsland, K.: Nurse scheduling with tabu search and strategic oscillation. Eur. J. Oper. Res. 106, 393–407 (1998)

    Article  MATH  Google Scholar 

  11. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Cover inequalities for 0–1 linear programs: computation. INFORMS J. Comput. 10, 427–437 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. http://www.hpcc.ttu.edu/index.php

  14. Ibaraki, T.: Complementary convex programming. J. Oper. Res. Soc. Japan 15, 138–160 (1972)

    MATH  MathSciNet  Google Scholar 

  15. Ibaraki, T.: The use of cuts in complementarity programming. Oper. Res. 21, 353–359 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ibaraki, T., Hasegawa, T., Teranaka, J., Iwasa, K.: The multiple-choice knapsack problem. J. Oper. Res. Soc. Japan 21, 59–93 (1978)

    MATH  Google Scholar 

  17. Jeroslow, R.G.: Cutting planes for complementarity constraints. SIAM J. Control Optim. 16, 56–62 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  18. Meyer, R.R.: Integer and mixed-integer programming models: general properties. J. Optim. Theory Appl. 16, 191–206 (1975)

    Article  MATH  Google Scholar 

  19. Meyer, R.R.: Mixed-integer minimization models for piecewise-linear functions of a single variable. Discrete Math. 16, 163–171 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mitchell, J.E., Pang, J.S., Yu, B.: Obtaining tighter relaxations of mathematical programs with complementarity constraints. J. Glob. Optim. (in press)

  21. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1988)

    Book  MATH  Google Scholar 

  22. Nguyen, T.T., Tawarmalani, M., Richard, J.P.P.: Convexification techniques for linear complementarity constraints. In: Günlük, O., Woeginger, G.J. (eds.) Integer Programming and Combinatorial (IPCO), Lecture Notes in Computer Science, vol. 6655, pp. 336–348. Springer, Berlin (2011)

  23. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3, 411–430 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Vielma, J.P., Ahmed, S., Nemhauser, G.L.: Mixed-integer models for nonseparable piecewise linear optimization: unifying framework and extensions. Oper. Res. 58, 303–315 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Progr. 128, 49–72 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Williams, H.P.: Model Building in Mathematical Programming, 4th edn. Wiley, New York (1999)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhao, M., de Farias, I.R.: Branch-and-cut for separable piecewise linear optimization: new inequalities and intersection with semi-continuous constraints. Math. Progr. A 141(1–2), 217–255 (2013)

    Google Scholar 

Download references

Acknowledgments

The authors acknowledge the High Performance Computing Center at Texas Tech University at Lubbock for providing resources that have contributed to the research results reported within this paper. URL: http://www.hpcc.ttu.edu. This research was partially supported by the Office of Naval Research (ONR) through grants N000140910332 and N000141310041. ONR’s support is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Texas Tech, Lubbock, USA

    I. R. de Farias Jr. & E. Kozyreff

  2. SAS, Cary, USA

    M. Zhao

Authors
  1. I. R. de Farias Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Kozyreff
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. R. de Farias Jr..

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Farias, I.R., Kozyreff, E. & Zhao, M. Branch-and-cut for complementarity-constrained optimization. Math. Prog. Comp. 6, 365–403 (2014). https://doi.org/10.1007/s12532-014-0070-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-014-0070-2

Keywords

Mathematics Subject Classification

Search

Navigation