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Solving Mathematical Programs with Equilibrium Constraints

Author

Listed:
  • Lei Guo

    (Shanghai Jiao Tong University
    University of Victoria)

  • Gui-Hua Lin

    (Shanghai University)

  • Jane J. Ye

    (University of Victoria)

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.

Suggested Citation

  • Lei Guo & Gui-Hua Lin & Jane J. Ye, 2015. "Solving Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 234-256, July.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:1:d:10.1007_s10957-014-0699-z
    DOI: 10.1007/s10957-014-0699-z
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    2. J. J. Ye & X. Y. Ye, 1997. "Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 977-997, November.
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    Cited by:

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    3. Badenbroek, Riley & Dahl, Joachim, 2020. "An Algorithm for Nonsymmetric Conic Optimization Inspired by MOSEK," Other publications TiSEM bcf7ef05-e4e6-4ce8-b2e9-6, Tilburg University, School of Economics and Management.
    4. Jean-Pierre Dussault & Mounir Haddou & Abdeslam Kadrani & Tangi Migot, 2020. "On Approximate Stationary Points of the Regularized Mathematical Program with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 504-522, August.
    5. Qingna Li & Zhen Li & Alain Zemkoho, 2022. "Bilevel hyperparameter optimization for support vector classification: theoretical analysis and a solution method," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 315-350, December.
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