IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v158y2013i1d10.1007_s10957-012-0228-x.html
   My bibliography  Save this article

Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints

Author

Listed:
  • Lei Guo

    (Dalian University of Technology)

  • Gui-Hua Lin

    (Shanghai University)

  • Jane J. Ye

    (University of Victoria)

Abstract

We study second-order optimality conditions for mathematical programs with equilibrium constraints (MPEC). Firstly, we improve some second-order optimality conditions for standard nonlinear programming problems using some newly discovered constraint qualifications in the literature, and apply them to MPEC. Then, we introduce some MPEC variants of these new constraint qualifications, which are all weaker than the MPEC linear independence constraint qualification, and derive several second-order optimality conditions for MPEC under the new MPEC constraint qualifications. Finally, we discuss the isolatedness of local minimizers for MPEC under very weak conditions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:1:d:10.1007_s10957-012-0228-x
    DOI: 10.1007/s10957-012-0228-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0228-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/s10957-012-0228-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. R. Andreani & C. E. Echagüe & M. L. Schuverdt, 2010. "Constant-Rank Condition and Second-Order Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 255-266, August.
    2. 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.
    3. 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.
    4. 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.
    5. R. Andreani & J. M. Martinez & M. L. Schuverdt, 2005. "On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 473-483, May.
    6. 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.
    7. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, 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. 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.
    2. Max Bucher & Alexandra Schwartz, 2018. "Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 383-410, August.
    3. 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.
    4. 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.
    5. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    6. Peng Zhang & Jin Zhang & Gui-Hua Lin & Xinmin Yang, 2019. "New Constraint Qualifications for S-Stationarity for MPEC with Nonsmooth Objective," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-16, April.
    7. 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.
    8. Lei Guo & Jin Zhang & Gui-Hua Lin, 2014. "New Results on Constraint Qualifications for Nonlinear Extremum Problems and Extensions," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 737-754, December.
    9. 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.
    10. Mengwei Xu & Jane J. Ye, 2020. "Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs," Journal of Global Optimization, Springer, vol. 78(1), pages 181-205, September.

    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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Patriksson, Michael, 2008. "On the applicability and solution of bilevel optimization models in transportation science: A study on the existence, stability and computation of optimal solutions to stochastic mathematical programs," Transportation Research Part B: Methodological, Elsevier, vol. 42(10), pages 843-860, December.
    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. 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.
    12. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    13. 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.
    14. 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.
    15. Kuang Bai & Yixia Song & Jin Zhang, 2023. "Second-Order Enhanced Optimality Conditions and Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1264-1284, September.
    16. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Discussion Paper 2004-25, Tilburg University, Center for Economic Research.
    17. 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.
    18. 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.
    19. Ashkan Mohammadi & Boris S. Mordukhovich & M. Ebrahim Sarabi, 2020. "Superlinear Convergence of the Sequential Quadratic Method in Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 731-758, September.
    20. 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.

    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:158:y:2013:i:1:d:10.1007_s10957-012-0228-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.