IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v55y2013i2p359-385.html
   My bibliography  Save this article

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

Author

Listed:
  • Jia Wu
  • Liwei Zhang
  • Yi Zhang

Abstract

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [ 40 ] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Jia Wu & Liwei Zhang & Yi Zhang, 2013. "A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations," Journal of Global Optimization, Springer, vol. 55(2), pages 359-385, February.
    • Handle: RePEc:spr:jglopt:v:55:y:2013:i:2:p:359-385
    DOI: 10.1007/s10898-012-9880-9
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-012-9880-9
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-012-9880-9?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. G.H. Lin & M. Fukushima, 2003. "Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 67-80, July.
    2. 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.
    3. Gemayqzel Bouza & Georg Still, 2007. "Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 467-483, May.
    4. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    5. Jia Chen & Yeol Cho & Jong Kim & Jun Li, 2011. "Multiobjective optimization problems with modified objective functions and cone constraints and applications," Journal of Global Optimization, Springer, vol. 49(1), pages 137-147, January.
    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. 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.
    2. Xide Zhu & Jin Zhang & Jinchuan Zhou & Xinmin Yang, 2019. "Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 521-540, May.

    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. 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.
    2. Wang, Guangmin & Gao, Ziyou & Xu, Meng & Sun, Huijun, 2014. "Joint link-based credit charging and road capacity improvement in continuous network design problem," Transportation Research Part A: Policy and Practice, Elsevier, vol. 67(C), pages 1-14.
    3. 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.
    4. 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.
    5. Axel Dreves & Christian Kanzow, 2011. "Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 50(1), pages 23-48, September.
    6. Flam, Sjur & Ruszczynski, A., 2006. "Computing Normalized Equilibria in Convex-Concave Games," Working Papers 2006:9, Lund University, Department of Economics.
    7. Giorgia Oggioni & Yves Smeers & Elisabetta Allevi & Siegfried Schaible, 2012. "A Generalized Nash Equilibrium Model of Market Coupling in the European Power System," Networks and Spatial Economics, Springer, vol. 12(4), pages 503-560, December.
    8. Alejandro Jofré & R. Terry Rockafellar & Roger J-B. Wets, 2007. "Variational Inequalities and Economic Equilibrium," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 32-50, February.
    9. Ciarcià, Carla & Daniele, Patrizia, 2016. "New existence theorems for quasi-variational inequalities and applications to financial models," European Journal of Operational Research, Elsevier, vol. 251(1), pages 288-299.
    10. Bin Chen & Nan-jing Huang, 2013. "Continuity of the solution mapping to parametric generalized vector equilibrium problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1515-1528, August.
    11. Contreras, Javier & Krawczyk, Jacek & Zuccollo, James, 2008. "The invisible polluter: Can regulators save consumer surplus?," MPRA Paper 9890, University Library of Munich, Germany.
    12. Ludovic A. Julien, 2024. "Noncooperative oligopoly equilibrium in markets with hierarchical competition," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 325-371, June.
    13. Jiang, Zhoutong & Lei, Chao & Ouyang, Yanfeng, 2020. "Optimal investment and management of shared bikes in a competitive market," Transportation Research Part B: Methodological, Elsevier, vol. 135(C), pages 143-155.
    14. Alexey Izmailov & Mikhail Solodov, 2014. "On error bounds and Newton-type methods for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 201-218, October.
    15. J. Contreras & J. B. Krawczyk & J. Zuccollo, 2016. "Economics of collective monitoring: a study of environmentally constrained electricity generators," Computational Management Science, Springer, vol. 13(3), pages 349-369, July.
    16. Benjamin F. Hobbs & J. S. Pang, 2007. "Nash-Cournot Equilibria in Electric Power Markets with Piecewise Linear Demand Functions and Joint Constraints," Operations Research, INFORMS, vol. 55(1), pages 113-127, February.
    17. Julien, Ludovic A., 2017. "On noncooperative oligopoly equilibrium in the multiple leader–follower game," European Journal of Operational Research, Elsevier, vol. 256(2), pages 650-662.
    18. Contreras, Javier & Krawczyk, Jacek & Zuccollo, James, 2008. "Can planners control competitive generators?," MPRA Paper 10395, University Library of Munich, Germany.
    19. Migot, Tangi & Cojocaru, Monica-G., 2020. "A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem," European Journal of Operational Research, Elsevier, vol. 283(3), pages 1136-1147.
    20. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.

    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:jglopt:v:55:y:2013:i:2:p:359-385. 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.