Skip to main content
SpringerLink
Log in

On solving Linear Complementarity Problems by DC programming and DCA

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

Abstract

In this paper, we consider four optimization models for solving the Linear Complementarity (LCP) Problems. They are all formulated as DC (Difference of Convex functions) programs for which the unified DC programming and DCA (DC Algorithms) are applied. The resulting DCA are simple: they consist of solving either successive linear programs, or successive convex quadratic programs, or simply the projection of points on \(\mathbb{R}_{+}^{2n}\). Numerical experiments on several test problems illustrate the efficiency of the proposed approaches in terms of the quality of the obtained solutions, the speed of convergence, and so on. Moreover, the comparative results with Lemke algorithm, a well known method for the LCP, show that DCA outperforms the Lemke method.

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

References

  1. Ahn, B.-H.: Iterative methods for linear complementarity problems with upper bounds on primary variables. Math. Program. 26, 295–315 (1983)

    Article  MATH  Google Scholar 

  2. Chen, X., Ye, Y.: On smoothing methods for the P 0 matrix linear complementarity problem. SIAM J. Optim. 11(2), 341–363 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  4. Fathi, Y.: Computational complexity of LCPs associated with positive definite symmetric matrices. Math. Program. 17, 335–344 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fernandes, L., Friedlander, A., Guedes, M.C., Judice, J.: Solution of a general linear complementarity problem using smooth optimization and its application to bilinear programming and LCP. Appl. Math. Optim. 43, 1–19 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fletcher, R.: Practical Methods of Optimization. Wiley, New York (1987). ISBN13:978-0471494638

    MATH  Google Scholar 

  7. Floudas, C.A., et al.: Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and Its Applications, vol. 33. Kluwer Academic, Dordrecht (1999). XV

    MATH  Google Scholar 

  8. Judice, J., Faustino, A.M., Ribeiro, I.M.: On the solution of NP-hard linear complementarity problems. Top 10(1), 125–145 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Geiger, C., Kanzow, C.: On the resolution of monotone complementarity problems. Comput. Optim. Appl. 5(2), 155–173 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Krause, N., Singer, Y.: Leveraging the margin more carefully In: Proceedings of the 21st International Conference on Machine Learning, ICML, 2004, Banff, Alberta, Canada, p. 63 (2004). ISBN:1-58113-828-5

    Chapter  Google Scholar 

  11. Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J. Glob. Optim. 11(3), 253–285 (1997)

    Article  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, Y., Shen, X., Doss, H.: Multicategory ψ-learning and support vector machine: computational tools. J. Comput. Graph. Stat. 14, 219–236 (2005)

    Article  MathSciNet  Google Scholar 

  14. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Sigma Series in Applied Mathematics, vol. 3. Heldermann, Berlin (1988),

    MATH  Google Scholar 

  15. Pardalos, P.M.: The linear complementarity problem. In: Gomez, S., Hennart, J.P. (eds.) Advances in Optimization and Numerical Analysis, pp. 39–49. Kluwer Academic, Norwell (1994).

    Google Scholar 

  16. Pardalos, P.M., Rosen, J.B.: Global optimization approach to the linear complementarity problem. SIAM J. Sci. Stat. Comput. 92, 341–353 (1988)

    Article  MathSciNet  Google Scholar 

  17. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pardalos, P.M., Ye, Y.: A class of linear complementarity problems solvable in polynomial time. Linear Algebra Appl. 152, 3–17 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Pham Dinh, T., Le Thi, H.A.: D.c. optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ronan, C., Fabian, S., Jason, W., Léon, B.: Trading convexity for scalability. In: Proceedings of the 23rd International Conference on Machine Learning, ICML 2006, Pittsburgh, Pennsylvania, pp. 201–208 (2006). ISBN:1-59593-383-2

    Google Scholar 

  22. Schnörr, C., Schüle, T., Weber, S.: Variational reconstruction with DC-programming. In: Herman, G.T., Kuba, A. (eds.) Advances in Discrete Tomography and Its Applications. Birkhäuser, Boston (2007)

    Google Scholar 

  23. Schüle, T., Schnörr, C., Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and d.c. programming. Discrete Appl. Math. 151, 229–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sherali, H., Krishnamurty, R., Al-Khayyal, F.: Enumeration approach for linear complementarity problems based on a reformulation-linearization technique. J. Optim. Theory Appl. 99, 481–507 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, Y.-B., Li, D.: A globally and locally superlinearly convergent non-interior-point algorithm for P0 LCPs. SIAM J. Optim. 13(4), 1195–1221 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Theoretical and Applied Computer Science (LITA), University of Paul Verlaine—Metz, Ile du Saulcy, 57045, Metz, France

    Hoai An Le Thi

  2. Laboratory of Modelling, Optimization & Operations Research, National Institute for Applied Sciences—Rouen, BP 08, Place Emile Blondel, 76131, Mont Saint Aignan 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. Tao Pham Dinh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hoai An Le Thi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Le Thi, H.A., Pham Dinh, T. On solving Linear Complementarity Problems by DC programming and DCA. Comput Optim Appl 50, 507–524 (2011). https://doi.org/10.1007/s10589-011-9398-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9398-y

Keywords

Search

Navigation