IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v124y2005i3d10.1007_s10957-004-1176-x.html
   My bibliography  Save this article

Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints

Author

Listed:
  • M.L. Flegel

    (University of Würzburg)

  • C. Kanzow

    (University of Würzburg)

Abstract

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:124:y:2005:i:3:d:10.1007_s10957-004-1176-x
    DOI: 10.1007/s10957-004-1176-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-004-1176-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-004-1176-x?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. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Peng Zhang & Jin Zhang & Gui-Hua Lin & Xinmin Yang, 2018. "Constraint Qualifications and Proper Pareto Optimality Conditions for Multiobjective Problems with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 763-782, March.
    2. Yi Zhang & Jia Wu & Liwei Zhang, 2015. "First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints," Journal of Global Optimization, Springer, vol. 63(2), pages 253-279, October.
    3. Arnav Ghosh & Balendu Bhooshan Upadhyay & I. M. Stancu-Minasian, 2023. "Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 11(17), pages 1-28, August.
    4. Savin Treanţă & Balendu Bhooshan Upadhyay & Arnav Ghosh & Kamsing Nonlaopon, 2022. "Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 10(19), pages 1-20, September.
    5. Balendu Bhooshan Upadhyay & Arnav Ghosh, 2023. "On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 1-35, October.
    6. 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.
    7. Patrick Mehlitz, 2020. "A comparison of solution approaches for the numerical treatment of or-constrained optimization problems," Computational Optimization and Applications, Springer, vol. 76(1), pages 233-275, May.
    8. 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.
    9. Gerd Wachsmuth, 2015. "Mathematical Programs with Complementarity Constraints in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 480-507, August.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. Monique Guignard, 2007. "En hommage à Joseph-Louis Lagrange et à Pierre Huard," Annals of Operations Research, Springer, vol. 149(1), pages 103-116, February.

    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. Feijoo, Felipe & Das, Tapas K., 2014. "Design of Pareto optimal CO2 cap-and-trade policies for deregulated electricity networks," Applied Energy, Elsevier, vol. 119(C), pages 371-383.
    2. J. S. Pang, 2007. "Partially B-Regular Optimization and Equilibrium Problems," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 687-699, August.
    3. 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.
    4. 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).
    5. 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.
    6. 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.
    7. Thai Doan Chuong, 2020. "Optimality conditions for nonsmooth multiobjective bilevel optimization problems," Annals of Operations Research, Springer, vol. 287(2), pages 617-642, April.
    8. Bhuwan Chandra Joshi, 2021. "Some Results on Mathematical Programs with Equilibrium Constraints," SN Operations Research Forum, Springer, vol. 2(4), pages 1-18, December.
    9. Elias S. Helou & Sandra A. Santos & Lucas E. A. Simões, 2020. "Analysis of a New Sequential Optimality Condition Applied to Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 433-447, May.
    10. 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.
    11. 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.
    12. 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.
    13. Yongchao Liu & Huifu Xu & Jane J. Ye, 2011. "Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 670-694, November.
    14. Zhang, Fang & Lu, Jian & Hu, Xiaojian & Meng, Qiang, 2023. "Integrated deployment of dedicated lane and roadside unit considering uncertain road capacity under the mixed-autonomy traffic environment," Transportation Research Part B: Methodological, Elsevier, vol. 174(C).
    15. Andreas Ehrenmann & Karsten Neuhoff, 2009. "A Comparison of Electricity Market Designs in Networks," Operations Research, INFORMS, vol. 57(2), pages 274-286, April.
    16. Gui-Hua Lin & Mei-Ju Luo & Jin Zhang, 2016. "Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 487-510, November.
    17. 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.
    18. Tao Tan & Yanyan Li & Xingsi Li, 2011. "A Smoothing Method for Zero–One Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 65-77, July.
    19. S. Dempe & S. Franke, 2016. "On the solution of convex bilevel optimization problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 685-703, April.
    20. A. F. Izmailov & M. V. Solodov, 2002. "The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 614-635, August.

    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:spr:joptap:v:124:y:2005:i:3:d:10.1007_s10957-004-1176-x. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.