IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v154y2012i1d10.1007_s10957-012-9992-x.html
   My bibliography  Save this article

Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints

Author

Listed:
  • Yan-Chao Liang

    (Dalian University of Technology)

  • Gui-Hua Lin

    (Dalian University of Technology)

Abstract

We consider the mathematical program with vertical complementarity constraints. We show that the min-max-min problems and the problems with max-min constraints can be reformulated as the above problem. As a complement of the work of Scheel and Scholtes in 2000, we derive the Mordukhovich-type stationarity conditions for the considered problem. We further reformulate various popular stationarity systems as nonlinear equations with simple constraints. A modified Levenberg–Marquardt method is employed to solve these constrained equations.

Suggested Citation

  • Yan-Chao Liang & Gui-Hua Lin, 2012. "Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 54-70, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-012-9992-x
    DOI: 10.1007/s10957-012-9992-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-9992-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/s10957-012-9992-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. E. Polak & J. O. Royset, 2003. "Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 421-457, December.
    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.
    3. Hui-juan Xiong & Bo Yu, 2010. "An aggregate deformation homotopy method for min-max-min problems with max-min constraints," Computational Optimization and Applications, Springer, vol. 47(3), pages 501-527, November.
    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. Stefan Scholtes, 2004. "Nonconvex Structures in Nonlinear Programming," Operations Research, INFORMS, vol. 52(3), pages 368-383, June.
    2. Hui-juan Xiong & Bo Yu, 2010. "An aggregate deformation homotopy method for min-max-min problems with max-min constraints," Computational Optimization and Applications, Springer, vol. 47(3), pages 501-527, November.
    3. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    4. B. Rustem & S. Žaković & P. Parpas, 2008. "Convergence of an Interior Point Algorithm for Continuous Minimax," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 87-103, January.
    5. Birbil, S.I. & Bouza, G. & Frenk, J.B.G. & Still, G.J., 2003. "Equilibrium Constrained Optimization Problems," Econometric Institute Research Papers ERS-2003-085-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    6. 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).
    7. Andreas Ehrenmann & Karsten Neuhoff, 2009. "A Comparison of Electricity Market Designs in Networks," Operations Research, INFORMS, vol. 57(2), pages 274-286, April.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Aram V. Arutyunov & Alexey F. Izmailov, 2005. "Sensitivity Analysis for Cone-Constrained Optimization Problems Under the Relaxed Constraint Qualifications," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 333-353, May.
    13. 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.
    14. 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.
    15. Pineda, Salvador & Boomsma, Trine K. & Wogrin, Sonja, 2018. "Renewable generation expansion under different support schemes: A stochastic equilibrium approach," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1086-1099.
    16. S. Dempe, 2012. "Comments on: Algorithms for linear programming with linear complementarity constraints," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 28-29, April.
    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. Alfred Auslender & Miguel A. Goberna & Marco A. López, 2009. "Penalty and Smoothing Methods for Convex Semi-Infinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 303-319, May.
    19. Birbil, S.I. & Bouza, G. & Frenk, J.B.G. & Still, G.J., 2003. "Equilibrium Constrained Optimization Problems," ERIM Report Series Research in Management ERS-2003-085-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    20. Grimm, Veronika & Martin, Alexander & Weibelzahl, Martin & Zöttl, Gregor, 2014. "Transmission and Generation Investment in Electricity Markets: The Effects of Market Splitting and Network Fee Regimes," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 460, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.

    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:154:y:2012:i:1:d:10.1007_s10957-012-9992-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.