IDEAS home Printed from https://ideas.repec.org/a/wsi/apjorx/v35y2018i01ns0217595918500082.html
   My bibliography  Save this article

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

Author

Listed:
  • Na Xu

    (School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China)

  • Xide Zhu

    (Faculty of Business Administration, Yokohama National University, Yokohama 240-8501, Japan)

  • Li-Ping Pang

    (School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China)

  • Jian Lv

    (School of Finance, Zhejiang University of Finance and Economics, Hangzhou 310018, P. R. China)

Abstract

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

Suggested Citation

  • Na Xu & Xide Zhu & Li-Ping Pang & Jian Lv, 2018. "Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(01), pages 1-20, February.
    • Handle: RePEc:wsi:apjorx:v:35:y:2018:i:01:n:s0217595918500082
    DOI: 10.1142/S0217595918500082
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0217595918500082
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0217595918500082?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Nguyen Huy Chieu & Gue Myung Lee, 2013. "A Relaxed Constant Positive Linear Dependence Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 11-32, July.
    2. J. V. Outrata, 1999. "Optimality Conditions for a Class of Mathematical Programs with Equilibrium Constraints," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 627-644, August.
    3. 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.
    4. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    5. 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.
    6. 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.
    7. Jane J. Ye & Jin Zhang, 2014. "Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 777-794, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. 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.
    3. Yogendra Pandey & S. K. Mishra, 2018. "Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators," Annals of Operations Research, Springer, vol. 269(1), pages 549-564, October.
    4. Alberto Ramos, 2019. "Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 566-591, November.
    5. Giandomenico Mastroeni & Letizia Pellegrini & Alberto Peretti, 2021. "Some numerical aspects on a method for solving linear problems with complementarity constraints," Working Papers 16/2021, University of Verona, Department of Economics.
    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.
    7. Hoai An Thi & Thi Minh Tam Nguyen & Tao Pham Dinh, 2023. "On solving difference of convex functions programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 163-197, September.
    8. Suhong Jiang & Jin Zhang & Caihua Chen & Guihua Lin, 2018. "Smoothing partial exact penalty splitting method for mathematical programs with equilibrium constraints," Journal of Global Optimization, Springer, vol. 70(1), pages 223-236, January.
    9. 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.
    10. Christian Kanzow & Alexandra Schwartz, 2014. "Convergence properties of the inexact Lin-Fukushima relaxation method for mathematical programs with complementarity constraints," Computational Optimization and Applications, Springer, vol. 59(1), pages 249-262, October.
    11. Yogendra Pandey & Shashi Kant Mishra, 2016. "Duality for Nonsmooth Optimization Problems with Equilibrium Constraints, Using Convexificators," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 694-707, November.
    12. Filippo Pecci & Edo Abraham & Ivan Stoianov, 2017. "Penalty and relaxation methods for the optimal placement and operation of control valves in water supply networks," Computational Optimization and Applications, Springer, vol. 67(1), pages 201-223, May.
    13. Winterfeld, Anton, 2008. "Application of general semi-infinite programming to lapidary cutting problems," European Journal of Operational Research, Elsevier, vol. 191(3), pages 838-854, December.
    14. Nguyen Huy Chieu & Gue Myung Lee, 2014. "Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and their Local Preservation Property," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 755-776, December.
    15. Daniel Ralph & Oliver Stein, 2011. "The C-Index: A New Stability Concept for Quadratic Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 504-526, August.
    16. Acuna, Jorge A. & Zayas-Castro, Jose L. & Feijoo, Felipe, 2022. "A bilevel Nash-in-Nash model for hospital mergers: A key to affordable care," Socio-Economic Planning Sciences, Elsevier, vol. 83(C).
    17. Hu, X. & Ralph, R., 2006. "Using EPECs to model bilevel games in restructured electricity markets with locational prices," Cambridge Working Papers in Economics 0619, Faculty of Economics, University of Cambridge.
    18. 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.
    19. Meng Xu & Guangmin Wang & Susan Grant-Muller & Ziyou Gao, 2017. "Joint road toll pricing and capacity development in discrete transport network design problem," Transportation, Springer, vol. 44(4), pages 731-752, July.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:apjorx:v:35:y:2018:i:01:n:s0217595918500082. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/apjor/apjor.shtml .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.