Skip to main content
SpringerLink
Log in

An enhanced logical benders approach for linear programs with complementarity constraints

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This work extends the logical benders approach for solving linear programs with complementarity constraints proposed by Hu et al. (SIAM J Optim 19(1):445–471, 2008) and Bai et al. (Comput Optim Appl 54(3):517–554, 2013). We develop a novel interpretation of the logical Benders method as a reversed branch-and-bound search, where the whole exploration procedure starts from the leaf nodes in an enumeration tree. This insight enables us to provide a new framework over which we can combine master problem and cut generation in a single process. It also allows us to diversify the search, leading computationally to stronger cuts. We also present an optimization-based sparsification process which makes the cut generation more efficient. Numerical results are presented to show the effectiveness of this enhanced method. Results are also extended to problems with more complementarity constraints, exceeding those that can be handled by the original method in the cited references.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Notation: \(N_n(0,1)\), refers to n i.i.d. draws from a standard normal distribution; \(U_n(a,b)\), n draws from a uniform distribution in the (a, b) interval, and \(DU(\{a_1,\ldots ,a_m\})\), a uniform draw from the set \(\{a_1,\ldots ,a_m\}\).

  2. We write “worst” within quotes because if this case happens, the cut would have sparsity at most 2, i.e., very sparse.

References

  1. Achterberg, T.: Constraint integer programming. Ph.D. Thesis, Technische University Berlin (2007)

  2. Alvarez, A.M., Louveaux, Q., Wehenkel, L.: A machine learning-based approximation of strong branching. INFORMS J. Comput. 29(1), 185–195 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, L., Mitchell, J.E., Pang, J.S.: On convex quadratic programs with linear complementarity constraints. Comput. Optim. Appl. 54(3), 517–554 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bard, J.F., Moore, J.T.: A branch and bound algorithm for the bilevel programming problem. SIAM J. Sci. Stat. Comput. 11(2), 281–292 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belotti, P., Bonami, P., Fischetti, M., Lodi, A., Monaci, M., Nogales-Gomez, A., Salvagnin, D.: On handling indicator constraints in mixed integer programming. Comput. Optim. Appl. 65(3), 545–566 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonami, P., Lodi, A., Tramontani, A., Wiese, S.: On mathematical programming with indicator constraints. Math. Program. 151, 191–223 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burdakov, O.P., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. 26(1), 397–425 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted \(\ell _1\) minimization. J. Fourier Anal. Appl. 14(5–6), 877–905 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Feng, M., Mitchell, J.E., Pang, J.S., Shen, X., Wächter, A.: Complementarity formulations of \(\ell _0\)-norm optimization problems. Pac. J. Optim. 14(2), 273–305 (2018)

    MathSciNet  Google Scholar 

  10. Fischer, T., Pfetsch, M.E.: Branch-and-cut for linear programs with overlapping SOS1 constraints. Math. Program. Comput. 10(1), 33–68 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fischetti, M., Monaci, M.: Backdoor branching. INFORMS J. Comput. 25(4), 693–700 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fischetti, M., Monaci, M.: Exploiting erraticism in search. Oper. Res. 62(1), 114–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optim. Methods Softw. 19(1), 15–40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation, Elsevier, pp. 89–134 (2008)

  15. Gomes, C.P., Selman, B., Kautz, H.A.: Boosting combinatorial search through randomization. In: Proceedings of AAAI/IAAI 1998, p. online. AAAI (1998)

  16. Gurobi Optimization: Inc.: Gurobi optimizer reference manual, 2015. URL: http://www.gurobi. com (2014)

  17. Hooker, J.N., Ottosson, G.: Logic-based Benders decomposition. Math. Program. 96(1), 33–60 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, J., Mitchell, J.E., Pang, J.S.: An LPCC approach to nonconvex quadratic programs. Math. Program. 133(1–2), 243–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, J., Mitchell, J.E., Pang, J.S., Bennett, K.P., Kunapuli, G.: On the global solution of linear programs with linear complementarity constraints. SIAM J. Optim. 19(1), 445–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, J., Mitchell, J.E., Pang, J.S., Yu, B.: On linear programs with linear complementarity constraints. J. Glob. Optim. 53(1), 29–51 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, J.: The effect of restarts on the efficiency of clause learning. In: Proceedings of IJCAI-07, pp. 2318–2323 (2007)

  22. Ibaraki, T.: Complementary programming. Oper. Res. 19(6), 1523–1529 (1971)

    Article  MATH  Google Scholar 

  23. Ibaraki, T.: The use of cuts in complementary programming. Oper. Res. 21(1), 353–359 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. IBM ILOG: Cplex. Book 12.2 User’s Manual (2010)

  25. Jeroslow, R.G.: Cutting-planes for complementarity constraints. SIAM J. Control Optim. 16(1), 56–62 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeroslow, R.G.: The polynomial hierarchy and a simple model for competitive analysis. Math. Program. 32(2), 146–164 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Karzan, F.K., Nemhauser, G.L., Savelsbergh, M.W.: Information-based branching schemes for binary linear mixed integer problems. Math. Program. Comput. 1(4), 249–293 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optim. 17(1), 52–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lodi, A., Zarpellon, G.: On learning and branching: a survey. TOP 25(2), 207–236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  31. Williams, R., Gomes, C.P., Selman, B.: Backdoors to typical case complexity. In: Proceedings of IJCAI-2003, pp. 1173–1178 (2003)

  32. Yu, B.: A branch and cut approach to linear programs with linear complementarity constraints. Ph.D. Thesis, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (2011)

  33. Yu, B., Mitchell, J.E., Pang, J.S.: Solving linear programs with complementarity constraints using branch-and-cut. Math. Program. Comput. (2018). https://doi.org/10.1007/s12532-018-0149-2

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Industrial, Universidad de Santiago de Chile, Av. Ecuador 3769, Santiago, Estación Central, Región Metropolitana, Santiago, Chile

    Francisco Jara-Moroni

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180-1590, USA

    John E. Mitchell

  3. Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    Jong-Shi Pang

  4. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 60208, USA

    Andreas Wächter

Authors
  1. Francisco Jara-Moroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John E. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jong-Shi Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andreas Wächter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Jara-Moroni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of Jara-Moroni was partially funded by the CONICYT PFCHA/DOCTORADO BECAS CHILE/2013-72140466 and the National Science Foundation under grant CMMI-1334639.

The work of Mitchell was funded by the National Science Foundation under grants CMMI-1334327 and DMS-1736326. The work of Pang was based on research supported by the Air Force Office of Scientific Research under grant FA9550-15-1-0126 and the National Science Foundation under grant CMMI-1402052.

The work of Wächter was funded by the National Science Foundation under grant CMMI-1334639.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jara-Moroni, F., Mitchell, J.E., Pang, JS. et al. An enhanced logical benders approach for linear programs with complementarity constraints. J Glob Optim 77, 687–714 (2020). https://doi.org/10.1007/s10898-020-00905-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00905-z

Keywords

Search

Navigation