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Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints

Author

Listed:
  • Lei Guo

    (East China University of Science and Technology)

  • Gaoxi Li

    (Chongqing Technology and Business University)

Abstract

We consider how to solve a class of non-Lipschitz mathematical programs with equilibrium constraints (MPEC) where the objective function involves a non-Lipschitz sparsity-inducing function and other functions are smooth. Solving the non-Lipschitz MPEC is highly challenging since the standard constraint qualifications fail due to the existence of equilibrium constraints and the subdifferential of the objective function is unbounded due to the existence of the non-Lipschitz function. On the one hand, for tackling the non-Lipschitzness of the objective function, we introduce a novel class of locally Lipschitz approximation functions that consolidate and unify a diverse range of existing smoothing techniques for the non-Lipschitz function. On the other hand, we use the Kanzow and Schwartz regularization scheme to approximate the equilibrium constraints since this regularization can preserve certain perpendicular structure as in equilibrium constraints, which can induce better convergence results. Then an approximation method is proposed for solving the non-Lipschitz MPEC and its convergence is established under weak conditions. In contrast with existing results, the proposed method can converge to a better stationary point under weaker qualification conditions. Finally, a computational study on the sparse solutions of linear complementarity problems is presented. The numerical results demonstrate the effectiveness of the proposed method.

Suggested Citation

  • Lei Guo & Gaoxi Li, 2024. "Approximation Methods for a Class of Non-Lipschitz Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1421-1445, September.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:3:d:10.1007_s10957-024-02475-6
    DOI: 10.1007/s10957-024-02475-6
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    References listed on IDEAS

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    1. Christian Kanzow & Alexandra Schwartz, 2015. "The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 253-275, February.
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    4. G.H. Lin & M. Fukushima, 2003. "New Relaxation Method for Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 81-116, July.
    5. H. Luo & X. Sun & Y. Xu & H. Wu, 2010. "On the convergence properties of modified augmented Lagrangian methods for mathematical programming with complementarity constraints," Journal of Global Optimization, Springer, vol. 46(2), pages 217-232, February.
    6. H. Z. Luo & X. L. Sun & Y. F. Xu, 2010. "Convergence Properties of Modified and Partially-Augmented Lagrangian Methods for Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 489-506, June.
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    8. Gemayqzel Bouza & Georg Still, 2007. "Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 467-483, May.
    9. Jane J. Ye, 2006. "Constraint Qualifications and KKT Conditions for Bilevel Programming Problems," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 811-824, November.
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