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A New Augmented Lagrangian Method for MPCCs—Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods

Author

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  • Lei Guo

    (School of Business, East China University of Science and Technology, Shanghai 200237, China)

  • Zhibin Deng

    (School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China)

Abstract

We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure; and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

Suggested Citation

  • Lei Guo & Zhibin Deng, 2022. "A New Augmented Lagrangian Method for MPCCs—Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods," Mathematics of Operations Research, INFORMS, vol. 47(2), pages 1229-1246, May.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:2:p:1229-1246
    DOI: 10.1287/moor.2021.1165
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    References listed on IDEAS

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    1. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
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    3. A. Izmailov & A. Pogosyan & M. Solodov, 2012. "Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints," Computational Optimization and Applications, Springer, vol. 51(1), pages 199-221, January.
    4. Lei Guo & Gui-Hua Lin, 2013. "Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 600-616, March.
    5. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
    6. X. M. Hu & D. Ralph, 2004. "Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 365-390, November.
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    Cited by:

    1. Yang Xu & Guyan Ni & Mengshi Zhang, 2024. "Bounds of the Solution Set to the Polynomial Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 146-164, October.
    2. Yan-Chao Liang & Yue-Wen Liu & Gui-Hua Lin & Xide Zhu, 2023. "New Constraint Qualifications for Mathematical Programs with Second-Order Cone Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 199(3), pages 1249-1280, December.
    3. Christian Kanzow & Patrick Mehlitz, 2022. "Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 624-646, November.
    4. 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.
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    6. Roberto Andreani & Mariana da Rosa & Leonardo D. Secchin, 2026. "A New Constant-Rank-Type Condition Related to MFCQ and Local Error Bounds," Journal of Optimization Theory and Applications, Springer, vol. 209(1), pages 1-26, April.

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