Skip to main content
SpringerLink
Log in

Solving Mathematical Programs with Equilibrium Constraints

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper aims at developing effective numerical methods for solving mathematical programs with equilibrium constraints. Due to the existence of complementarity constraints, the usual constraint qualifications do not hold at any feasible point, and there are various stationarity concepts such as Clarke, Mordukhovich, and strong stationarities that are specially defined for mathematical programs with equilibrium constraints. However, since these stationarity systems contain some unknown index sets, there has been no numerical method for solving them directly. In this paper, we remove the unknown index sets from these stationarity systems successfully, and reformulate them as smooth equations with box constraints. We further present a modified Levenberg–Marquardt method for solving these constrained equations. We show that, under some weak local error bound conditions, the method is locally and superlinearly convergent. Furthermore, we give some sufficient conditions for local error bounds, and show that these conditions are not very stringent by a number of examples.

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

Notes

  1. In fact, when we chose the initial penalty parameter to be 1, the numerical results obtained are not satisfactory.

References

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

    Book  Google Scholar 

  2. Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers, Boston (1998)

    Book  MATH  Google Scholar 

  3. Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Guo, L., Lin, G.H., Ye, J.J.: Stability analysis for parametric mathematical programs with geometric constraints and its applications. SIAM J. Optim. 22, 1151–1176 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Guo, L., Lin, G.H., Ye, J.J., Zhang, J.: Sensitivity analysis for parametric mathematical programs with equilibrium constraints. SIAM J. Optim. 24, 1206–1237 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fukushima, M., Lin, G.H.: Smoothing methods for mathematical programs with equilibrium constraints. In: Proceedings of the ICKS’04, pp. 206–213. IEEE Computer Society Press, Los Alamitos (2004)

  10. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming Theory and Algorithms. Wiley, New York (1993)

    MATH  Google Scholar 

  11. Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 172, 375–397 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fan, J.Y., Yuan, Y.X.: On the quadratic convergence of the Levenberg–Marquardt method without nonsingularity assumption. Computing 74, 23–39 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Behling, R., Fischer, A.: A unified local convergence analysis of inexact constrained Levenberg–Marquardt methods. Optim. Lett. 6, 927–940 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wu, Z., Ye, J.J.: First-order and second-order conditions for error bounds. SIAM J. Optim. 14, 621–645 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hoffman, A.J.: On approximate solutions of systems of linear equalities. J. Res. Nat. Bur. Stand. 49, 263–265 (1952)

    Article  Google Scholar 

  16. Jourani, A.: Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory Appl. 81, 533–548 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21, 314–332 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guo, L., Ye, J.J., Zhang, J.: Mathematical programs with geometric constraints in Banach spaces: enhanced optimality, exact penalty, and sensitivity. SIAM J. Optim. 23, 2295–2319 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17, 259–286 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Huang, X.X., Yang, X.Q., Teo, K.L.: Partial augmented Lagrangian method and mathematical programs with complementarity constraints. J. Global Optim. 35, 235–254 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yang, X.Q., Huang, X.X.: Lower-order penalty methods for mathematical programs with complementarity constraints. Optim. Methods Softw. 19, 693–720 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first author’s work was supported by the NSFC Grant (No. 11401379) and the China Postdoctoral Science Foundation (No. 2014M550237). The second author’s work was supported in part by the NSFC Grant (No. 11431004) and the Innovation Program of Shanghai Municipal Education Commission. The third author’s work was supported in part by the NSERC. The authors are grateful to the anonymous referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Sino-US Global Logistics Institute, Shanghai Jiao Tong University, Shanghai, 200030, China

    Lei Guo

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada

    Lei Guo & Jane J. Ye

  3. School of Management, Shanghai University, Shanghai, 200444, China

    Gui-Hua Lin

Authors
  1. Lei Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gui-Hua Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jane J. Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Hua Lin.

Additional information

Communicated by Xiaoqi Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, L., Lin, GH. & Ye, J.J. Solving Mathematical Programs with Equilibrium Constraints. J Optim Theory Appl 166, 234–256 (2015). https://doi.org/10.1007/s10957-014-0699-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0699-z

Keywords

Mathematics Subject Classification

Search

Navigation