Skip to main content
SpringerLink
Log in

On solving difference of convex functions programs with linear complementarity constraints

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We address a large class of Mathematical Programs with Linear Complementarity Constraints which minimizes a continuously differentiable DC function (Difference of Convex functions) on a set defined by linear constraints and linear complementarity constraints, named Difference of Convex functions programs with Linear Complementarity Constraints. Using exact penalty techniques, we reformulate it, via four penalty functions, as standard Difference of Convex functions programs. The difference of convex functions algorithm (DCA), an efficient approach in nonconvex programming framework, is then developed to solve the resulting problems. Two particular cases are considered: quadratic problems with linear complementarity constraints and asymmetric eigenvalue complementarity problems. Numerical experiments are performed on several benchmark data, and the results show the effectiveness and the superiority of the proposed approaches comparing with some standard methods.

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

Similar content being viewed by others

Data availability

The data that supports the findings of this study is available from the corresponding author upon request.

Notes

  1. https://math.nist.gov/MatrixMarket/matrices.html.

  2. https://www.artelys.com/en/optimization-tools/knitro.

  3. https://neos-server.org/neos/solvers.

References

  1. Audet, C., Savard, G., Zghal, W.: New branch-and-cut algorithm for bilevel linear programming. J. Optim. Theory Appl. 134, 353–370 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Bard, J.F.: Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68, 371–378 (1991)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85, 107–134 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Fukushima, M., Pang, J.S.: Convergence of a smoothing continuation method for mathematical problems with complementarity constraints. In: Théra, M., Tichatschke, R. (eds.) Ill-posed Variational Problems and Regularization Techniques. Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 99–101. Springer, Berlin (1999)

    Google Scholar 

  7. Fukushima, M., Luo, Z.Q., Pang, J.S.: A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints. Comput. Optim. Appl. 10, 5–34 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Hau Luu, H.: Techniques avancées d’apprentissage automatique bas

  9. Hobbs, B.F., Metzler, C.B., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15(2), 638–645 (2000)

    Google Scholar 

  10. Hu, X., Ralph, D.: Convergence of a penalty method for mathematical programming with equilibrium constraints. J. Optim. Theory Appl. 123, 365–390 (2004)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Huang, X.X., Yang, X.Q., Zhu, D.L.: A sequential smooth penalization approach to mathematical programs with complementarity constraints. Numer. Func. Anal. Opt. 27(1), 71–98 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Jara-Moroni, F., Pang, J.S., Wächter, A.: A study of the difference-of-convex approach for solving linear programs with complementarity constraints. Math. Program., Special Issue: DC Programming - Theory, Algorithms and Applications 169(1), 221–254 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Jiang, H., Ralph, D.: Extension of quasi-newton methods to mathematical programs with complementarity constraints. Comput. Optim. Appl. 25, 123–150 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Júdice, J.J.: Optimization with linear complementarity constraints. Pesquisa Operacional 34, 559–584 (2014)

    Google Scholar 

  16. Júdice, J.J., Sherali, H.D., Ribeiro, I.M., Faustino, A.M.: A complementarity-based partitioning and disjunctive cut algorithm for mathematical programming problems with equilibrium constraints. J. Glob. Optim. 36, 89–114 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Júdice, J.J., Sherali, H.D., Ribeiro, I.M.: The eigenvalue complementarity problem. Comput. Optim. Appl. 37, 139–156 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Kadrani, A., Dussault, J.P., Benchakroun, A.: A new regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 20(1), 78–103 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. SIAM J. Optim. 23(2), 770–798 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Kunapuli, G., Bennett, K.P., Hu, J., Pang, J.S.: Classification model selection via bilevel programming. Optim. Methods Softw. 23, 475–489 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Le Thi, H.A.: DC Programming and DCA: http://www.lita.univ-lorraine.fr/~lethi/index.php/en/research/dc-programming-and-dca.html(homepage) (2005)

  22. Le Thi, H.A., Pham Dinh, T.: Large-scale molecular optimization from distance matrices by a DC optimization approach. SIAM J. Optim. 14(1), 11–114 (2003)

    MATH  Google Scholar 

  23. Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Le Thi, H.A., Pham Dinh, T.: On solving linear complementarity problems by DC programming and DCA. Comput. Optim. Appl. 50, 507–524 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Le Thi, H.A., Pham Dinh, T.: DC programming and DCA: thirty years of developments. Math. Program., Special Issue: DC Programming - Theory, Algorithms and Applications 169(1), 5–68 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Le Thi, H.A., Pham Dinh, T., Nguyen Canh, N., Nguyen, V.T.: DC programming techniques for solving a class of nonlinear bilevel programs. J. Glob. Optim. 44, 313–337 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Le Thi, H.A., Moeini, M., Pham Dinh, T., Júdice, J.J.: A DC programming approach for solving the symmetric eigenvalue complementarity problem. Comput. Optim. Appl. 51(3), 1097–1117 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Le Thi, H.A., Pham Dinh, T., Huynh, V.N.: Exact penalty and error bounds in DC programming. J. Glob. Optim. 52, 509–535 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: DC Programming and DCA for general DC Programs. In: Do van, T., Le Thi, H.A., Nguyen, N.T. (eds.) Advanced Computational Methods for Knowledge Engineering, Advances in Intelligent Systems and Computing, pp. 15–35. Springer, Cham (2014)

  30. Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: Convergence analysis of difference-of-convex algorithm with subanalytic data. J. Optim. Theory Appl. 179(1), 103–126 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Le Thi, H.A., Huynh, V.N., Pham Dinh, T., Luu, H.: Stochastic difference-of-convex functions algorithms for nonconvex programming. SIAM J. Optim. 32(3), 2263–2293 (2022)

    MathSciNet  MATH  Google Scholar 

  32. Leyffer, S.: MacMPEC http://wiki.mcs.anl.gov/leyffer/index.php/MacMPEC (webpage) (2000)

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

    MathSciNet  MATH  Google Scholar 

  34. Li, Y., Tan, T., Li, X.: A log-exponential smoothing method for mathematical programs with complementarity constraints. Appl. Math. Comput. 218, 5900–5909 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Lin, G.H., Fukushima, M.: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 118(1), 67–80 (2003)

    MathSciNet  MATH  Google Scholar 

  36. Lin, G.H., Fukushima, M.: Modified relaxation scheme for mathematical programs with complementarity constraints. Ann. Oper. Res. 133, 63–84 (2005)

    MathSciNet  MATH  Google Scholar 

  37. Liu, X., Sun, J.: Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints. Math. Program. 101, 231–261 (2004)

    MathSciNet  MATH  Google Scholar 

  38. Liu, G.S., Ye, J.J.: Merit-function piecewise SQP algorithm for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 135, 623–641 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Liu, G.S., Zhang, J.Z.: A new branch and bound algorithm for solving quadratic programs with linear complementarity constraints. J. Comput. Appl. Math. 146, 77–87 (2002)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  41. Mangasarian, O.L., Pang, J.S.: Exact penalty for mathematical programs with linear complementarity constraints. Optimization 42, 1–8 (1997)

    MathSciNet  MATH  Google Scholar 

  42. Niu, Y.S., Pham Dinh, T., Le Thi, H.A., Júdice, J.J.: Efficient DC programming approaches for the asymmetric eigenvalue complementarity problem. Optim. Methods Softw. 28(4), 812–829 (2013)

    MathSciNet  MATH  Google Scholar 

  43. Niu, Y.S., Júdice, J., Le Thi, H.A., Pham Dinh, T.: Solving the quadratic eigenvalue complementarity problem by DC programming. In: Le Thi, H.A., Pham Dinh, T., Nguyen, N.T. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, Advances in Intelligent Systems and Computing, vol. 359, pp. 203–214 (2015)

  44. Pang, J.S., Leyffer, S.: On the global minimization of the value-at-risk. Optim. Methods Softw. 19, 611–631 (2004)

    MathSciNet  MATH  Google Scholar 

  45. Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: theory, algorithms and applications. Acta Math. Vietnam 22(1), 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  46. Pham Dinh, T., Le Thi, H.A.: A DC Optimization algorithm for solving the trust region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)

    MathSciNet  MATH  Google Scholar 

  47. Pham Dinh, T., Le Thi, H.A.: Recent advances in DC programming and DCA. In: Nguyen, N.T., Le Thi, H.A. (eds.) Transactions on Computational Collective Intelligence XIII. Lecture Notes in Computer Science, vol. 8342, pp. 1–37. Springer, Berlin (2014)

    Google Scholar 

  48. Pieper, H.: Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity market. Ph.D. thesis, Standford University (2001)

  49. Raghunathan, A.U., Biegler, L.T.: Barrier methods for mathematical programs with complementarity constraints (MPCCs). Technical report, Carnegie Mellon University, Department of Chemical Engineering, Pittsburgh, PA (2002)

  50. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationary, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)

    MathSciNet  MATH  Google Scholar 

  51. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001)

    MathSciNet  MATH  Google Scholar 

  52. Scholtes, S., Stöhr, M.: Exact penalization of mathematical programs with equilibrium constraints. SIAM J. Control. Optim. 37(2), 617–652 (1999)

    MathSciNet  MATH  Google Scholar 

  53. Steffensen, S., Ulbrich, M.: A new relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 20(5), 2504–2539 (2010)

    MathSciNet  MATH  Google Scholar 

  54. Yu, B.: A branch and cut approach to linear programs with linear complementarity constraints. Ph.D. thesis, Rensselaer Polytechnic Institute (2011)

Download references

Acknowledgements

The authors are grateful to the Associate Editor and two referees for offering many constructive comments that have improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Université de Lorraine, LGIPM, Metz, 57000, France

    Hoai An Le Thi

  2. Institut Universitaire de France (IUF), Paris, France

    Hoai An Le Thi

  3. Faculty of Fundamental Science, Posts and Telecommunications Institute of Technology, Km10, Nguyen Trai, Ha Dong District, Hanoi, Vietnam

    Thi Minh Tam Nguyen

  4. Laboratory of Mathematics (LMI), National Institute for Applied Sciences, Avenue de l’Université-BP 8, 76801, Saint-Etienne-du-Rouvray Cedex, France

    Tao Pham Dinh

Authors
  1. Hoai An Le Thi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thi Minh Tam Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Pham Dinh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tao Pham Dinh.

Ethics declarations

Conflict of interest

All authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (zip 23951 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le Thi, H.A., Nguyen, T.M.T. & Dinh, T.P. On solving difference of convex functions programs with linear complementarity constraints. Comput Optim Appl 86, 163–197 (2023). https://doi.org/10.1007/s10589-023-00487-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-023-00487-y

Keywords

Mathematics Subject Classification

Search

Navigation